JPRS ID: 10417 TRANSLATION THEORETICAL AND EXPERIMENTAL INVESTIGATIONS OF SURFACE AND INTERNAL WAVES ED. BY B.A. NELEPO

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. JpRS L/ 10417
26 March 19.82
Translation
= THEORETICAL AND EXPERIMENTAL
INVESTIGATIONS OF SURFACE AND INTERNAL YNAVES
Ed. by
B.A. Nelepo~
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JPRS L/10~17
26 March 1982
- T'HEORETICAL AND EXPERIMENTAL INVESTIGATIONS
OF SURFACE A~JD INTERNAL Wf~VES
5,evastopol' TEORETICHESKTI'~' ~C EKSPERIMENT~IL'NYYE ISSLEDOVANIYA POVERK-
HNOSTNYRH T VNUTRENNIKH ~i10~~T in Russian 1980 pp 1-202
[Translation of "Theoretical and Experimental Investigations o= Sur-
fa.ce and Internal Waves", edited by B. A. Nelepo, academic:an Ukrain-
ian Academy of Sciences, Morskoy gidrofizicheskiy institut AN USSR (MGI
AN USSR), 300 copies, 202 pages]
Annotation 1
Some Droblems in Wave Diffraction and Scattering by Spatial Inl:omo~
geneities in Ocean and Atmosphere..
~ (I. T. Selezov) 3
Nonstationary Three-Dimensional Waves in a Flaw of Hom~geneous F1uid
With Velocity Shear
(A. M. Suvorov, et al.) 14
Effect of Current Velocity Shear on Amplitudes of Waves Generated in
Homogeneous Fiuid by Mo~-Cng Pressure Region
(V. F. Sannikov, L. V. Cherkesov) 22
Unsteady Three-Dimensional Waves in a Flow of Homogeneous Fluid With
Velocity Shear
(A. M. Suvorov, A. N.. Tananayev) 31
Compv~ations of Distributions of Orbital Velocities of Wind Waves
(L. A. Korneva, V. P. Liverdi) 41
Effect of Geometry of Pressure Digturbances on Characteristics of
Capillarty-GravitationalTdaves
(V. V~, Trepachev) 51
- a - ~
[I - USSR - E F'VUn]
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Unsteady Flexural-Gravitational Waves Fiom Impulse Disturbar.ces
- Under Ice Compression Conditions
(A. Ye. Bukatov, A. A. Yaroshenko) 61
Investigation of Tsunami Waves in the Neighborhood of Iturup
Island
(R. A. Yaroshenya) 69
Ge nerar.ion of Seismic Waves by Tsunami Waves Propagating in Ocean
With Uneven Bottom
(I. V. Lavrenov) 73
Evolution of Axi.symmetric Disturbances of Viscous Fluid
(B. Ye. Sergeyevskiy) 83
Effect of Viscosity and Beta-Effect on Generation of Long Wav~s
in Ocean by Atmospheric GJaves
(V. F. Ivanov) 93 ,
Generation of Long Waves in Ocean Over Local Bottom Rise by
Atmospheric Waves
- (V. F. Ivanov) 100
Influence of Fluid Viscosity on Wave Resistance to System of
Normal Stresses Distributed in Segment
(L. G. Yeremenko) 110
Long Surface cr.d Internal Waves Generated by NonaxisyL,~etric
~ Initial Disturbances
(S. F. Ibtsenko, B. Yu. Sergeyevskiy) 116
Internal Waves From Initial Disturbances in a Two-Iayer Fluid
(A. A. Novik) 128
Investigation of Effect of Vertical Density Structure on Internal
Wave s
(S. M. Khartiyev, L. V. Cherkesov) 135
Effect of Fine Stra~tification on Internal Waves Generated by
Periodic Atmospheric Disturbances
(S. M. Khartiyev, L. V. Ch~rkesov) 143
Evaluation of Possibla Values of Parameters o~ Internal Waves in
South Polar Fron t Zone
(N. P. Bulgakov, R. A. Yaroshenya) 151
Variability of Energy Density of Internal Waves With Depth in an
= Inhom~geneously Stratified Ocean
(V. Z. Dykman, A. A. Slepyshev) ~59
- b -
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Effect of Surface Film on Natural Oscil].ations of Free Boundary of
F, .~i d
(T. M. Pogorelova) 1E~6
E�fect ~f Viscosity on Dissipation of Internal Waves .
(N. P. Levkov) 173
Free Oscillations of a Stratified Viscous Fluid
(N. P. Levkov) 180
~
~
~
, - c -
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~
[Text] Annotation. This collectian of articles contains the results of theoretical
_ and experimental investigations of surface, barotropic and fnternal waves arising
in the ocean under the influence of periodic, moving and pulsed disturbances.
These disturbances are wind, pressure formations, barotropic waves and submarine
eartY~quakes. The articles discuss the depeiidence of the characteristics of wave pro-
cesses on the parameters of the operative fc~rces, sea medium and bottom relief.
The collection is intended for specialists in the field of marine physics and hy-
drodynamics.
Preface. The Second Seminar of Directors and Participants in the Interdepartmental
Project "VolnaQ1 ["Wave"] was held in Sevastopol' during the period 26-30 November
� 1979 at the Marine Hydrophysical Institute of th~ Ukrainian Academy of Sciences.
The theme of the seminar was: "Theoretical and Experimental Investigations of Sur-
face and Internal Waves in the Ocean.47
The organizers o� this meeting of scientists were the Scientific Council of the
USSR State Committee on Science and Technology on the Problem "Study of the Oceans
and Seas and Use of Their Resources, the Commission on the Problem of the World
Ucean of the USSR Academy of Sciences Presidi.um and the Marine Hydrophysical Inst-
itute of the Ukrainiar~ Academy of Sciences as the coordinating organization of
the "Volna." pro~ ect .
The seminar was attended by specialists of the Marine Hydrophysical Institute of
the Ukrainian Academy of Sciences, the Institute of Oceanology of the USSR Academy
of Sciences, Acoustics Institute, Moscow State University, liydromechanics Znstitute
of the Ukrainian Academy of Sciences, Central Scientific Research Institute imeni
A. N. Krylov, Soyuzmorniiproyekt, Institute uf Appiied Physics of the USSR Academy
of Sciences, Northern Caucasus Scientific Center of Institutes of Higher Educa-
' tion,,Siberian Department of the USSR Aeademy of Sciences, Far Eastern Scientific
1 ~
i
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Cen.ter of the USSR Academ5� of Sciences, State Committee on Hydrometeorology and
Environmental Monitoring of the USSR Council of Mir~isters and a number ~f other
organizations.
More than 70 scientific repurts were presented and discussed. These dealt with
the present-day status of theoretical and experimental investigations of wave
processes in the ocean and timely directions in work in this field during 1981-
1985. There was a sc~~ntific discussion of a broad range of questions relating to
study of wind waves, mathematical modeling of the processes of generation and
propagation of surface and internal waves in the open ocean and in the coastal
zone, experimental investigation of internal waves under in situ and laboratory
conditions.
The work of the seminar transpired in plenary and section meetings in two sec-
tions: surface and internal waves. The participants in the meeting expressed an
interest in holding a scientific conference in 1980 and the publication of rs
specialized collection of articles.
The collection of articles contains presentations of studies presented at the sem-
inar and devoted to theoretical and experimental investigations of surface and in-
ternal waves generated in the ocean by periodic, pulsed and moving disturbances.
The first section includes articles which examine surface and long waves arising
in a barotropic ocean. The diffraction of waves by spatial inhomogeneities of bot-
tom relief is studied; the influence of the velocity shear of sea currents on
three-dimensional waves generateri by a moving pressure region is investigated; com-
putations of the distributions of orbital velocities of wind waves are given; the
role o~ the ice cover in the process of propagat~.on of flexural-gravitational
waves is analyzed. In the case of long waves a study is made of the generation of
seismic waves during the movement of tsunamis in a basin of variable depth, tsu-
nami waves are analyzed on the basis of observational data in the neighborhood of
Iturup Islazid, evolution of axisyuu~etric d~sturbances is studied, and ~he influ-
ence of viscosity and the ~6 -effect on waves generated by periodic disturban.ces
in a basin of variable depth is investigated.
The second section of this collection includes articles which examine internal
waves. Long-period spatial internal. waves arising from nonaxisymmetric initial
disturbances in the absence of a flow are analyzed; nonstationary short-period plane
internal waves in ttie presence of a flow with a sharp change in velocity are stud-
ied; the distorting influence of fine stratification on free and forced waves is
investigated; an evaluation of the possible values of elements of internal waves
in the zone of the southern polar front is given; the variability of the energy
density of internal waves with depth is considered; the influence of viscosity
on wave processes in the stratif ied ocean is analyzed.
2
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F( ~
SURFACE WAVES
SOME PROBLEMS IN WAVE DIFFRACTION AND SCATTERING BY SPATIAL INHOMOGENEITIES IN
OCEAN AND ATMOSPHERE
Sevastopol' TEORETICHESKIYE I EKSPERIMENTAL'NYYE ISSLEDOVANIYA POVERI~HNOSTNYKH I
VNUTRENNIKH VULN in Russian 1980 l,manuscript received 7 Jun 80) pp 7-18
[Article by I. T. Selezov]
[Text] Abstract: The problems of the theory of
waves are considered when axisymmetric in-
- ~ fiomogeneities are present. Some precise
solutions are found and the methods of ~en-
eralized power series and appro~~imation of
- solutions by polynumials are set forth.
The diffraction of waves in the broad sense is the deviation of wave movements
from the laws of geometrical optics. This phenomenon is generated when the medi~im
- contains some inhomogeneities localized or distributed in space.
As is well known, the application of rigorous methods of ma.thematical physics to
solution of the problems of wave propagation, diffraction and scattering is lim-
ited and belongs to cJ_assical problems. Recently approximate analytical and numer-
ical methods are being developed [1, 4, 6-9, 11-15].
l. Introduction. The classical problem of the theory of nonstationary diffraction
is formulated in the following way [5]. Assume that some closed region R with
the boundary ~ R is stipulated in the regic~n Q, oriented in a rectangular Cartes-
ian coordinate system x, y, z and a plane wave is propagated along the Ox-axis (Fig.
1). Then the diffraction problem is reduced to the Cauchy problem or the boundary-
value pxoblem for an equation in partial derivatives (or system of equations) in
tr~e Q region, whose coefficients experienr,e discontinuities of the first kind on
the d R surface. The ~ R surface is characterized by a curvilinear coordinate sys-
- tem.
We will assume that the field in the external region Q is described by the wave
equation for the function ~(x, y, z, t)
Z (1.1)
_ ~ ~ . ~ y 0 : . . . .
~ Co ~ - ~ ~
. cZ dt~ . . . .
3
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~ , ~ .
y ~ - � ,
Q . . .
R
. x
dR r . � ~ ~ ~ ,
Fig. 1.
The ~ function is. represented in the form of the sum of ti~e incident ~i and the
scattered ~3 fields ~i + ~s, where ~i(x,0,0,t) = F(x + ct).
We will assume that at the time t= 0 the leading edge of the wave touches the ~ R
surface. This is equivalent to the following:�for the sought-for function ~ and
its derivative t the following initial conditions are stipulated
~ I t=0 a du I~~o ~ p~ . y`~H ~ro - x ~F ~x+ct ) . (~.2>
- at
- The boundary-value conditions in the simplest variant have the form of Dirichlet
or Neumann conditions
a~ ~ D . ~i.3)
_ . . . . ~ ~ aR ' d ~ ( a~e
In addition, the solution mu~t satisfy the attenuation condition at fnfinity.
~
I H(r)
N "
� I
~ �ns
i
6.
Fig. 2.
After the substitution'of ~i +~S into (1.1)-(1.3) we obtain a homogeneous
equation relative to ~S, since [~i satisfies the wave equation,and inhomogeneous
houndary-value conditions, since C~ i is a stipulated function.
Scattered and incident fields.in diffraction problems are described in differ:nt
coordinate systems. Accc;edingly, one of the fields must be restruct~:red into
another coordinate system, which is not a simple groblem.
llepending on the properties of the R region, which in diffraction theory is cal?-
ed an inhomogeneity, obstacle, scatterer or diffracting body, it is possible to
dia.*.inguish three cases: ideally reflecting or translucent body, the field does
- not enter into the R region; transparent object whose properties are d~fferent
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from the surrounding medium, but are constant, that is, in the R region are not
dependent on space coordinates; an arbitrary inhomogeneity whose properties in
the R region are dependent on the space coordinates. In the two latter cases, in
addition to the external problem, it is also.necesr~ary to solve the internal prob-
lem.
The first case includes: absolutely rigid body in an acoustic field, ideally con-
ducting in an electromagnetic field and a vertical cylindez (junction) protruding
above the free surface in the field of surface gravitational waves. As an example
for the second and third cases we can use an underwater projection with a constant
or variable depth (Fig. 2) �in the field of surface gravitational waves.
If the function ~ is represented in the form of a monochromatic travelling wave
yc _ ~o e i (h~+G/t)~ ~h _ "!tj . '
~ (1.4)
we ~btain the stationary diffraction problem. It is assumed that - 00 G tG oo. The
sought-for function, in accordance with (1.4) can be cited in the form .
. ~ i~f _ . (1.5)
Cx,Y, Z, t~ ='.Lj CX, y,1~ P � .
The stationary diffraction problem is formulated on the basis of (1.1), (1.5) in
the form of a boundary-value problem for the Helmhcltz equation
(v ? + ) y = 0. (1. 6)
N
As before the solution is represented by the sum ~ ~i +~1~z. The boundary-value
conditions are similar for d R. In addition, in order to ensure uniqueness of the
solution we introduce the radiation condition and the condition of limitation at
_ infinity the Sommerfeld conditions. They have the form in three- and two-dimen-
sional cases
~
~~m r(a~- ~k~~o, ~~~~~a~-~,~y , e~~~-o.
r--~ dr ~ i'-�' r-��
L
2. Formulations of some problems. We wi~.l examine an axisy~netric inhomogeneity
whose center is matched with the origin of cylindrical (r, B, z) and spherical (r,
e, ~i') coordinate systems: f= f(r), 0 0 for which the integrand in (2) is nonnegative; Xm =
1, if (-1)m(sin Bm xl - cos Bm yl) > 0 and xm = 0 in the opposit.e case. The computa-
tions were made for the function
. f(~' r` ~ ~ eXP _ XZ y 1 ' f J)= ~ PXP ~ ~ai ju~ - Z (B . ( 3 )
2 1 I
, J Z~Ia,B L 1a, 18, J Z
'ihe integrals (2) have the following characteristics which must be taken into ac-
- count in the computations: infinite upper limit, the integrand can be unlimited
_ with r= rp and rapidly oscillate with large xl and yl values. We will first eval-
uate the residual term of the integral, rejectable with replacement of the infinite
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upper limit by a finite interval
_ , �o
rt cos (r coa B~ + y,si~ B�~] X~ dr I~ f rfdr~ (4)
B? I-�t~'~rctfir z z 6-I-a+x rcth j r
. i 1 e �~~s r :
~ � '
27I -1-a +~r~ 1 ctk Z az
where a2 = max (al, bl)/H. Selecting the upper limit r2 adequately large, it is
possible to make expression (4) as small as desired. The singularity of the inte-
grand at the point r= r~ is eliminated by the substitution r= rp + u2. The com~-
putation of the integrals (2) in the case of high xl and yl values requires the
use of quadrature formulas of the Philon type j4] or a great number of nodes in
the Gauss or Simpson formulas. With the use of the last integrals (2,) it is pos-
sible for different xl and yl to compute the quadrature formulas without scaling
of the weights. In this artticle we have used the Gauss formulas with the nutnber
of nodes being n= 40 [4]. With a parallel or~anization of the calculations the
computation of the rise of the free surface at 3,600 points required only 5 min-
utes on an M-220 electronic computer. The computationa were made for the follow-
ing values of the parameters of the problem:
x~ ~ 5,5, ~y, ~ c 3~>a,/H = B,~H ~~~1, .
(5)
a E~ 0~2; 0,2 8 E~0; 0~6, ~E ~0,25; 4,75~ .
In this case there did not have to be a further breakdown of the integration in-
terval for achieving a relative accuracy of 10-2-10-3.
2. Figure 1 gives a general idea concerning the influence of the shear of the
transverse component of current velocity on the wave trace behind the moving
pressure region. The values of the lines of equal deviation ~ from the undisturb-
ed level, shown in the figure, were normalized to the maximum z value in the re-
gion 0~ xl ~ 5.5; -2~ yl S 2. The current velocity vector turns to the left with
depth, in the direction of a negative direction of the y-axis. The extents of the
conspicuous parts of the corresponding crests and troughs (level lines f0.25) in
the right and lef t parts of the figure differ by a factor of approximately 2.5.
As follows from the results of the preceding study [3], with an increase in the
,B parameter the wave crests and troughs in the right part of the wave trace are
displaced toward the Ox-axis, but never pass beyond it. The computations indicated
that the maximum values of the Z deviations from the.undisturbed level are approx-
imately equal to the left and right of the Ox-axis for all values of the parameters
_ (5). Thus, the transverse current velocity shear can lead to substantial changes
in the phase configuration of the trace.
~ Figure 2 makes possible a more detailed tracing of changes in the wave trace oc-
curring with a chznge in the parameter b of shear of the transverse component of
current velocity. As the scale unit along the z-axis we took the maximum value
- 7~ (xl, yl) in the region 2~ xl~ 0 the Z~ extrema are displaced to the left with an increase in b and the
distances between adjacent extrema decrease.
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Table 1
Maximum Deviations of Fluid Surface From Undisturhed Level
- ~ B 0 0,1 Q,Z O,s o,4
0,25 15,18 18,18 18,20 18,20 15,~0
0~80 2I~42 21,80 21,94 21,97 :1,98
I,00 ~ 28,08 28,80 28,82 28,99 29,~5
1,50 SS,94 37,19 38,31 39,95 38,48
1,78 38,22 87,46 38,82 39,64 39,34
2,00 88,38 59,61 40,85 41,98 42,49
_ 2,25 37,37 8T,86 36,75 41,2? 43,43
2~50 84,63 35~17 38~08 57,~3 99,84
9~00 7,.5,30 25,98 28,25 32~28 35,62
4,00 ?,28 7,88 9~86 13~74 20,14
4~7~ 1~38 I,5S 2,26 4,10 ;,~,02
Table 2
Joint.Effect of Influeace of Shears o� Longitudinal and Transverse Components
- of Current Velocity on t~laximum~ Z~ Values
...6 O1 0
a - .
x ~.),2 -0~1 0,1 0~2 ~-0,2 -0~1 0,1 0~2 ,
- O,ZS ;5~13 1;.~16 15~21 15,24 18,14 15,f7 15,22 18,28
0,80 22,"�7 21~73 21~86 Z1~92 21,T7 21~88 22,03 22,16
1,OC ~28,12 28,3'I 80,84 32,41 28,36 28~81 29,SF 32~7
1,25 :~~;~74 33,~4 85,61 34~36 31,29 92~$8 36~82 36~08
1~5Q :'~;u 38~04 34,78 38~01 87,15 39,83 33,92 36,49
- 1,75 ~1,21 ;,7~02 ST,47 36~39 41,28 37~93 58,76 87~8&
2,00 ~~,58 ~'~,88 36,24 33,83 40~?5 41,42 87,85 33,48
2~25 42,14 41 ~47 34~11 30~27 43,88 42,82 34,44 31,11
2,50 42~Bc: 39,27 30,81 29~39 48,6T 38,24 32,18 26,93
3,00 90,ti? 31 ~92 19~89 13~88 40,89 84,78 21 ~76 14~53
4.00 24~44 1S~42 3,6Z 1~22 28~80 1T,77 4,67 1,75
- 4,78 12,18 S~OZ 0~32 - 15.24 8~83 0~65 0~09
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An analysis of the results of computations shows that the transverse velocity
shear, directed in the negative direction of the y-axis, elongates the wave crests
- in the region y~ 0 and turns them toward the Ox-axis (the angle between the crest
~tnd t:~e Ox-axis b~~omes less). In the region y= 0 the tratisverse velocity shear
s~ortens the wave crests and increases the angles between the crests and the Ox-
axis. The lengths of the waves increase in the region y> 0 and decrease in the
region y~ 0. With an ~ncrease in the b parameter the amplitudes of the waves
~ increase.
3. We will make an analysis of the dependence of the maximum deviations of the free
surface on the values of the parameters a, b, J~ . The maximiun ~ j i values were de-
termined for the region 2~ xi< 5, -1.5~ yl~ 1.5. The max ~ Z ~v~lues are given in
the tables with an accuracy to the factor p0/(2trp Hc2
Table 1 gives the results of maxll Icomputations for a= 0 and a s~ries of ~Cand b
values. The author of [3] gave the critical value of the parameter X=(1 + a-
b2)'1, with transition through which there is a change in the composition of the
~ wave trace. For a= 0 and 0~ b< 0.4 the critical ~ value falls in the interval 1
~ 3~< 1.2. It is interesting that with a fixed value of the b parameter the maximum
~deviations are attained with ~2 N 2, that is, greater than the critical
value. An analysis of the results of computations cited in T~ible 1 indicatee: that
for b~ 0.4 and aP~bP2the changes in the maximum deviations of the free surface do
not exceed 10-12~6. The influence of the transverse shear of current velocity is
greater with large df values (or small velocities of movement of the region of
pressures relative to the free surface). For example, with dC= 2 the change in
the maxl7~ ~value is 10.7% with a change in b 0-t7.4 and for 4.75 the maxllr~val-
. ue with b= 0.4 is almost six times greater than the corresponding value with b=
0. In addition, with a decrease in 2fthe changes in the maxll~~values decrease
with a change in the b parameter. With an increase in the b parameter for all
~P there is also an increase in the maximum deviations of the free surface from
the undisturbed level.
We will examine the joint effect of the influence of shears of the longitudinal
and transverse current velocity components on wave amplitudes. The results of
computations for b= 0.1, 0.2 and a number of values of the a and ~P parameters are
given in Table 2. The critical values of the ~ parameter here fall in the inter-
val 0.84~d~< 1.32. An analysis of the results of computations shows that with al-
lowance for the shear of the longitudinal component of current velocity the prin-
cipal effect of presence of a shear of the transverse component is reflected in
an increase in the amplitude of waves. The influence of longitudinal shear is
more varied. With small aC va.lues with an increase in the a parameter there is
an increase in the max~T ~ values; with large af values the picture is the reverse.
A chan~e in a is reflected to a greater degree in the case of great The
position of the maximum max ~1.~~values (with respect to is dependent on the
a value to a greater degree than on b. With a decrease in a there is an increase
in ~at which maxl~ ~attains its greatest value. For example, with b= 0.1 and
a=-0.2 the greatest maxll,~value is attained with 2.5 and with this same b
value and a= 0.1 with J~ = 1.75. ~de also note that these ~ values are greater
than the corresponding ~ values. ~
28
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Summary
The intensity of shear of the transverse component of current velocity vpZ exerts
a substantial influence on the phase configuration of the wave trace. With an in-
crease in ivO~ the wave crests from one side of the wave trace (syuunetrical with
vpZ = 0) become longer, and on the other side shorter. In this case the wave
crests are turned relative to the pressure epicenter in the direction of rotation
of the current velocity vector with depth (Fig. 1).
The maximum displacement of the free surface is an increasing function of the shear
modulus of the transverse component of current velocity IvOzl.
The greatest displacements of the free surface are attained with velocities C
of movement of disturbances less than the velocities of propagation of long waves
(critical).
In the case of small (in comparison with the critical values) velocities of move-
ment of disturbances the maximum displacements of the free surface increase with
- an increase in u~Z, the shear of the longitudinal component of cu:rent velocity,
and decrease with large C values.
The influence of shear of current velocity on the amplitudes of waves is most con-
spicuous in the case of small velocities of movement of disturbances. For example,
for a C velocity half as great as the critical value the wave amplitudes can~change
by a factor of 20 or more with a change in the shear parameters in the intervals
indicated in (5). In addition, the changes in maximum amplitudes of waves do not
exceed 15% for velocities of movement of disturbances greater than the critical
values.
BIBLIOGRAPHY
1. Cherkesov, L. V., NEUSTANOVIVSHIYESYA VOLNY (Unsteady Waves), Kiev, "Naukova
Dumka," 1970, 196 pages.
2. Suvorov, A. M., Tananayev, A. N. and Cherkesov, L. V., "Nonstationary Spatial
- Waves in a Flow of a Homogeneous Fluid With Velocity Shear," POVEF.I~iNOSTTTYYE
I VNUTRENNIYE VOLNY (Surface and Internal Waves), Sevastopol', Izd. MGI AN
Ukrainskoy SSR, pp 14-21, 1979.
3. Sannikov, V. F., "Ship Waves in a Homogeneous Sea With a Linear Current Velo-
city Profile With Depth," POVERKHNOSTNYYE I VNUTRENNIYE VOLNY, Sevastopol',
Izd. MGI AN Ukrainskoy SSR, pp 22-31.
4. Abramovits, M. and Stigan, I., SPRAVOCI~IIK PO SPETSIAL'NYM FUNKTSIYAM S FOR-
MULAMI, GRAFIKAMI I TABLITSAMI (Handbook on Special Functions With Formulas,
Graphs and Tables), Moscow, "Nauka," pp 673-720, 1979.
5. Kelvin (W. Thomson), "On the Waves Produced by a Single Impulse in Water of
Any Depth or in Dispersive Medium," PROC. ROY. SOC. LONDON, Ser. A, Vol 42,
PP 80-85, 1887.
29 .
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6. Havelock, T., "The Propagation of Groups of Waves in Dispersive Medium With
Application to Waves on Water Produced by a Travelling Disturbance," PROC.
ROY. SOC., LONDON, Ser. A, Vol 81, No 549, pp 398-430, 1908.
y. 'Kolberg, F., "Untersuchung des Wellenwiderstandes von Schiffen auf flachen
Wasser gleichformig scherender Grundstromung," ZEITSCHRIFT ANGE[J. MATH.,
B 39, H 7/8, 253-279, 1959.
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UNSTEADY THREE-DIMENSIONAL WAVES IN A FLOW OF HOMOGENEOUS FLUID WITH VELOCITY S~iEAR
Sevastopol' TEORETICHESKIYE I EKSPERIMENTAL'NYYE ISSLEDOVANIYA POVERKHNOSTTIYKH I
VNUTRENNIKH VOLN in Russian 1980 (mai~uscript received 28 Feb 80) pp 36-44
[Article by A. M. Suvorov and A. N. Tananayev]'
[Text] Abstract: A study was made of the process of de-
- velopment of three-dimensional surface waves
generated by moving atmospheric pressure dis-
turbances. Flow velocity is a piecewise-linear
function of the vertical coordinate, which makes
it possible, with a sufficient degree of accur-
- acy, to approximate both the direction and velo-
city modulus of real currents in the ocean. The
article gives a method for the analysis of three-
dimensional waves in an N-layer (with respect to
current velocity) aea. The special case with N=
1 is examined.
The theory of ship waves is one of the complex branches of modern hydrodynar~ics of
the sea [1-4], and at the same time is quite important f.~r practical applications.
Emphasis has been on study of wave movement in a medium at rest in an undisturbed
state or in a fluid flow having a velocity constant in depth. Recently interest has
been shown in investigation of the influence of the vertical structure of currents
on the parameters of three-dimensional waves [5, 6, 13]. In these studies current
velocity was assumed to be a linear function of depth. However, observations show
[7, 9J that one of the characteristic features of horizontal currents in the ocean
is a quite complex dependence of their velocity and direction on depth. In this
article we investigate unsteady three--dimensional surface waves for a piecewise-
linear vertical profile of flow.velocity making it possible with an adequate de-
gree of accuracy to approximate real currents in the ocean. .
1. We will examine the flow of an ideal incompressible homogeneous fluid of con- '
stant depth unbounded in horizontal directions (see Fig. 1). Her~ z=-h�(j = 0.1,
h0 = 0) the vertical coordinates of the points at which the modu~us and
direction of the velocity vector of the horizontal current are stipulated (for
example, on the basis of oceanic measurement data). In the intervals between these
points the components of the current velocity vectpr along the xl and yl axes were
approxim~ted by linear functions of depth
31
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U~- ~I~.cl, ~-d~1+ ~ . '
;
(1.1) i
, j
� Z i
;
~ ~
i ~ ~ y
. ,v y
X~ ~ ~ - ---hl-
~
~ ~ 1 yZ
. ~ U _ -h:
~ ~ -
~
~
~
~
~
~
~ �
- h,y,-,
~ - .
i ~ ,
/
, , y y
~
~ ~ N I 'y
i ~~N
- f ?1 i n *Tr
f~//
i
f~'' 'f ~ ~
. ~
Fig. 1.
It is also assumed that
_ _ ' - y'
~ i ~ ' j ~
/ % with z~ - ~t~ (~~J,..., N-!)
~
the derivatives of current velocity at these points can have discontinuities of
the first kind.
. Assume that a region of atmospheric pressure dlsturbances in the form
Pct=Po7C(Xi+Ut, r~+Y t) ~ (1.27
movin~ at a cons.tant velocity is imparted to tTie surface of a fluid flow at the
time t = 0.
Within the framework of linear theory we will investigate the process of develop-
ment of three-dimensional waves generated by. disturbances (1.2) and described by
the following system of equations, boundary and initial conditions, ~written with
(1.1) taken into account,
32
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~~L~ cc~ ~B ~)�~-,o~~~ , QCL~ y~+dw~)--p,,~. pL~ ~--p~Z~ (i.3)
u.~ + y.r ~ w. ~ p~ C co r~
Later it was found that the experimental data on ~(hp~To) do not correspond to the
assumption of a nondependence of this conditional distribution on Tp [2]. This
is the reason for the discrepancy betwean the experimental distributions and the
theoretical distributions, obtained with use of the theorem of multiplication of ~
laws for independent values. In our recent investigations in the Black SEa the
basis for obtaining different distributions is the initial two-dimensional ni]�
- table having i rows for the hp values and j columns for the T~ values. If it 3s �
taken into account that _
~ ~ ho ~ T ~i.i , 9~ ~T ~ - R~J ' .
~~~~al~ a~~aT
c .
it can then be shown that the theorem of multiplication of laws on the basis of
the data in such a table is satisf ied in general form (1), since
_ . _ _ -
R~,j
_ ~~~o,T~ ~ J aitoaT ~ .
~
It is possible to proceed from the main system (h0, T~) and the two-dimensional
distribution ~(h0, T~) corresponding to it to systems of values related to hp,
T0.
In this study we will attempt to find ni~(hp/T~, Tp), ~(hp/Tp, and from them
~(v~) and ~(v), where v=?1 h/T, and v~ = hp/T~ = v/v*; v* = n h T.
In order to obtain an ni~(h0, T~) table we used two two-dimensional tables (one of
them included 4079 pairs of h~, T~ values, taken from wave records of the Black
Sea, whereas the other included 8116 pairs of h~, T~ values oUtained on voyages
- in the Atlantic Ocean [3]). The h~ and T~ values were used in computing vp = hp/Tp
and the oUtained orbital velocity value V~ is assigned the number of cases of ni~
(h~, Tp) found in this same box in the table. Then a statistical table nii (vQ, TO)
is obtained for equal intervals ~v~ and L1T~. In our computations we stipulated
0 v~ = 0.4 and Q T~ = 0.2. From the ni~(v0, Tp) values we find all the other types
42
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~ ,
~ . -
a,
~
~ ~
E-i c0 ~
ti m~ ti g g$~~j
F u t ~ 00 ~ O p O
_ ~ O O C O O O p O O O p"
O
P '
v
~ ~ ~ ~ O
N 1 ~
q cd C
O N
~ Cn
~
a u
~ ~ C~ t~ m er
1~.1 fA � � w . .
y ~ N 00 N
~i 1+
- A O
w
~
L'. O : '~Y m : .
.o v ~ pp, O
t~ ~C
~rl G! N
~ t~ OD 00 M ~ 0~
V p ~ ~ ~ N ~ O
~ ~
N ~ rr
o~ o~ c~ or~ cw
~ ~ CD t~ ~ W pj O
O tA
E-~ ~ ~ .
O ma ~eDS
Oa~.~ l~ O�., '~0' W~' 0~ 'Q' ~ N m~ W = d! : ..w
w w w ~ ~ w w ~ O O � O ^y
~ 0 ~ O r"i ~ 0~ O ~ V 0
~
~
L; N
i?
r~l A ^ ~ ~l ~
~ ~ r., ~ C? ~ ~ ~
c~d W 00 ~ f0 N~ m~~O ! w N M G~ ~ C p: 0~ ' w �
rl G 0 O~t~ ^'pp00 ~yOmO pr~.,.p~0~ ^
~r1 . N O t~ O m.. QO
_ ~ ~ v v �
N ~
~
A ~
c~o c~d ti ti^Q~ W C3 aD ~ oQ ~ O~ :cD ~ G~~ c+~ O 2
O t~- O~~ ~~~0 m0 00~ 1~0000000 p
N ~ v
G". �rl
_ C1 ^ .
- 8 N ~r ~ ~r ~ ^ p~ ^
p ffi O � o O
AW p O~N +W~!`~ ~N ~ ~:~~W~~ON C7
3 ti O~NQNO00~OM~Ovp'..p~-:OO.~C
v v v.rv
W l`~ ~A
O : p ~ r�~ ( , * ~R, .
O ~ O ^ k
~n y
v
~ t~o
`~O u
~ ~N ~ m O ~ GO N m O Q: 00 N~
~ ~ ~O O O CV c~t o~ ~ c~ ~ aJ`
A ' u
i
t
43
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of distributions. Table 1 gives the probahility of the t~o-dimensional distribu-
tion computed using the formula
l RlJ
~o~ 0 . ~3)
~
The density of the conditional distribution of orbital velocity for a fixed period
TD is computed for each column of the two-dimensional table _
/~i vo, T . � .
P~v~~�T,~ _ (4)
~ R~~ d Vp
The conditional distribution of the period for a fixed value can be computed for
- each line of the two-dimensional table
y~T~~o~= - .
~ R~~ d T
~
- The total values in the table columns and rows are used in computing the marginal
distributions of orbital velocity ~(vp) and period ~(T~) respectively
~ R~J
~~r~ _ ~ ' Y~~o~=~ � (5)
. e r ~ R~~ d v
!
The ~ eX(TD) values are given in Table 1 in the last additional row; the ~eX~~O)
values are given in the last column.
The ~(v0 IT~) values can also be found using the theorem of multiplication of
laws since they are equal to
7 Il~s ~~~--T~
These values are given in Table 1 in parentheses for some Tp values. It can be
seen that ~(v~ (T~) differs from ~(V~). Naturally, the statistical characteris-
tics of the conditional distributions will also differ from the characteristics of
the marginal di~tributions, that is, �aill not be constant, but will change in de-
pendence on the fixed Tp value. An analysis of ~(v~,~T~) and the statistical
characteristics of this distribution (such as mean v0, the dispersion 6vp2, the
variation coefficient C~~~) was made for the Black Sea and the Atlantic Ocean
(Table 2). The changes in thes.e characteristics are shown in Fig. 1. It can be seen
that the considered statistics (statistical characteristics) are not constant but
are dependent on T~. With TD = 0.85 for the Black Sea and with T~ = 0.7 for the
ocean there is a maximum on the v~(TD) curve. For 6 there is a maximum with a
value T~ = 0.6 for the two considered regions. The variation coefficient CV~~
changes smoothly from large values in the case of small T0, attains values 0.52
with T~ = 0.8 and then decreases to 0.35 with T~ = 2. We investigated the experi-
mental marginal distrihution ~(v~) _~(v)v* for tlie Black Sea and ~(v~) for the
- 44
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ocean. These distributions are characterized hy the following statiatics: v~ _
1.03, 6 ~o2= ~�27 for the B1ack Sea; v~ ~ 1.04, 6 yfl = Q.26 for the ocean. It can
be seen that the marginal distriTiutions t~p(v~) for both water regtons are charac-
terized by one and the same variation coefficient Cp~ = Q.52 and the value vp =
1.03-1.04. Tfie greatest difference ~(vQ fT~) from tp ~v~) is observed for waves
of short and long periods and the difference is less clearly expressed for periods
close to the average.
~ Table 2
Statistical Characteristics of Conditional Distribution ~(v0 IT~) as a Function
of Different TD
Blaclc Sea Oceau
r ~ ~~yy 6y 4 ~v 6t .
0~2 0,79 0~75 0~26 0,94 0,?E 0,90
0,4 Q,85 0,85 D,94 1,10 0,68 O~S6
0,8 1~10 0,66 a,36 1~15 O~b4 0~86
0,8 1,18 Or49 0~32 1~14 0.46 O,S1
1,0 1~14 0,44 ~ 0~25 1,08 0~44 ~ O,Zl
1,2 1,02 0~40 0,17 0,95 0~42 0,14
1,4 0~89 0~38 O~lI 0~82 0,38 0.10
I~B 0~76 0,98 0,07 0,11 0~37 0~0'1
1,8 0,67 0,88 O~OS 0~64 O,SB 0,06
2,0 0~63 0~34 0~04 0~82 0,84 0,06
2~2 0~60 0,84 0,08 0,60 0,38 0~05
Now we will examine in greater detail the marginal distribu~ion ~(hp/T~) _~(v~)
because we must obtain from it the distribution ~~u)v, where v= 37 h/T is the true
orbital velocity. For this we use the notation ~h/T = v*, ?ih/T = v. Then h0/T~
= v/v* = v~. Thus, the distribution ~(v~) is the distribution ~(v/v*). For the
linearly related arguments v and v/v* there will be a correlation of the distribu-
tions themselves [1] in the form ~~~~l
~ / ~
Ldith transformation from the v/v* values to v/v we will assume that v* =,JB v. The
~ value is found from the experimental distribution ~(v~) _~(v/v*), computing
the first moment of the distribution. It is equal to
45
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. _ - - - -
~
C ~l ~~)~SC~I~~) ~ c~/~*) ~~I.)�
~ � c6)
vo , ~ _ ` ~
/ ~
!,0 ~ ~
1 ~
' ~
~
~
.
OJ
G~K . ~
0,~ ~
~
~
0,4 ~ ,
r�~ /
Qj ~ \
~
p - ~ .
0 1 2 ;
Fig. 1. Statistical characteristics v ITO, CV~~ITQ, ( IT~ of the conditional dis-
tribuCion ~(v~ (T~): 1) Black Sea; 24 ocean.
Since the ~(v~) values are expressed in the numerical data (Table 1), computations
of the first moment were made by the numerical integration method, where use was
made of the method for computing the statistical characteristics by the approach of
grouping of values by classes [4].
It was therefore found that for the distribution ~(v~) (v/v*) = 1.03-1.04 for the
Black Sea and the ocean respectively. It can be assumed that 1.04. The scaling
of cp (v~) into ~(v)v ~ras carried out on the assumption that c~ (v)v =~(v~)~ with
the argument v/v = v/~ v*.
Tab1e 3(columns 2,3) gave experimental data for the Black Sea and ocean and in Fig.
2 it can be seen that the experimental distributions of the orbital velocities of
the two water regions coincide quite well.
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Figure 3 shows the integral curves
Y 4
f(~l~) _ ~ ~ (v/v')dC~l~~
- whose values are expressed in percent (sn-called buaranteed probability);using
these curves it is possible to determine the ratio of wave velocity to mean or-
- bital velocity for waves of stipulated guaranteed probability.
Table 3
Experimental Generalized Distribution ~(v)v for Black Sea and Ocean and Its
Computation From ~(hp, nh) and ~(Tp, nT)
Cornauted ~ Y~ V
d~~ Black Sea Ocean /lr ��4,
~1~ =S ~ =2 R
~,2 0,20 0,16 0,89 0~42
0,4 0,44 0,48 0,66 0~86
0,6 0,73 0,71 0,80 . 0~T8 .
0~8 0,95 0,90 D,80 0,76
1,0 O,t~O 0,88 0~67 0~66
1,2 0,$6 0,78 0,50 O~S9
1,4 0,42 0~48 0,36 0~38
I,8 0,29 ~ 0,32 0,26 0,26
1,8 0,20 0,21 0~18 O,ZO
2,0 0,13 0~14 0~18 0~14
~ ,2 . 0,07 0 ~07 0,09 0,10
2,4 0,04 0,04 0,0? 0,07
2,6 0,02 0,02 0,05 0,05
2.8 0~01 0,01 0,04 O~C4
8~0 0,01 0~01 0,09 0,09
For a comparison of the experimental material with the theoretical scheme we made
computations of the theoretical value ~(vp) from ~(hp, T~) in two variants:
~,~ho, 0~=6/'t ~Z ~ r~~~~koTozexp[,r~?/"vt-~J~3/of~~
~'r(ko. a ~=8 r1\L3)~ 4\~/ho o3~xPl
/"1(?l'o2-~y~~jp~, � ~8)
In both cases we will assume that ~(hp, Tp) _~(~p)~ (T~), that is, that ~(hp)
c3nd ~(Tp) are independent and their approximation was taken in the form of a
Weibull distribution cP(x) = nAxpn-1 exp [-AxnJ, where A=(~~n + 1/n). In the
first variant nh = 2,.nT = 3, in the second nh = 2, nT = 4. These nh and nT values
; were selected on the basis of the experimental values of the variation coefficients
~ which we obtained C~T~ = 0.31 and Cphp = 0.52 for the Black Sea and CVT~ = 0.34,
' ~~Vhp - 0.52 for the ocean [3, 5]. The determined ~1(hp, Tp) and ~2(h~, Tp) val-
~ ues were used in forming two variants of two-dimensional c~(hp, Tp) tables.
~
i 47
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Computations were made for all the theoretical distributions.
~l(~~~ ~ ~
~o
~
~ .
~ ~ 2
~ ~ L
~s ~ ,
_ ~ ~
~
,
~
,
� T- 2 ~ ~I~
Fig. 2. Distributions ~(v)v: 1) based on experimental data for the Blaclc Sea;
2) on the basis of oceanic data; 3) computations of ~f (v)v from ~(hp, nh) ~(T~, ~
nT) with nh = 2, nT = 3 and 4) with nh = 2, nT = 4.
f (~l%
lOd ~ .
.
~ 2
~
~
~
~ 50 ~
. ~ ~7 ~
~
0
- ! ~ 9 Vo
Fig. 3. Integral distribution guaranteed probability obtained from the corres-
ponding curves in Fig. 2.
The computed distributions ~(v~) are given in Table 3(columns�4, 5 and in Fig.
2). In comparing the different curves in Figures 2 and 3 we note that the two dif-
ferent variants with respect to nT do not give a substantial difference in the
values of the approximating computation curves ~(vp). A somewhat greater dis-
- crepancy is observed when they are compared with experimental data: the most prob-
able ~(v)v values (Fig. 2) do not coincide; for vp ~ 1 the theoretical guaranteed
48
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probability curves are less than the experimental values (Fig. 3). Such a differ-
ence between the experimental and computed distributions should be expected since
the fundamental assumption of theoretical computations on the equality CQ(v~ ITD)
_~(vp) does not correspond to the dependence which we obtained between the con-
ditional distributions and their statistical parameters on Tp, as was mentioned
in the first part of the study.
It was demonstrated earlier in [3] on the basis of an analysis of ~(b and now
in this article in the example of ~(v),that the simplified theorem of multiplica-
tion of laws (2) is not applicable to computations of the original distribution of
waves; it can be considered when ~(h~ (T~) is unknown. In order to refine such
computations it is necessary to apply the theorem of multiplication of laws in
its general form (1), which agrees with the method which we used in computing dif-
ferent types of distribution on the basis of experimental two-dimensional tables.
Accordingly, experimental curves and tables characterizing different distributions
of orbital velocities are recommended.
- ~ ,
V~ V
1 ~101 .
~ � ~ ' _
6 - ~ .
/
5 ~
/ ~
/ ~
4 i
p I 2 ~ t~ v
_ 0 1 ~ S�!03 x/VI
Fig. 4. Changes in V*/V values at different stages in wave development (t/V): 1) on
the basis of experimental data for the Black Sea; 2) for ocean; 3) our computations
using the V. V. Shuleykin diagrams method [6].
As tiie distribution parameter ~(v), changing in dependenc~ on the stage of wave
clevelopment, it is possible to use v*/V. Its numerical value can be precomputed
since it is related to h and T, whose values are determined on the basis of the
experimental data in [S1. Figure 4 shows the dependence of v*/V on t/V, obtained
from the dependence of h/V2 and T/V on t/V. As the latter we rook the experimental
dependences [3] for the Black Sea and ocean and the prognostic expressions in the
V. V. Shuleykin method [6]. It can be seen that there is a specific difference in
the course of development of waves in the Black Sea and ocean.
In conclusion we will generalize the scheme for computing orbital velocities. On
the basis of the wave-forming factors V, t and x we determine t/V and x/V2. If
on the x-scale (Fig. 4) x/V2 < t/V, the waves must be considered as not developed,
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but rather developing [7] and the V*/V values are determined in this case on the
basis of t/V. Computations of well-developed waves are made from the dependence
of v~/V on x/V2. Since the wind is stipulated, the numerical value of v,~/V and V
makes it possible to determine the numerical value v* = Vv*/V. Then we determine
v~+~8 v*. On the basis of known v and the S~(v)v values we determine ~(v), giving
the probability of all values of specific v= Vv/V different from v. Using Fig. 3
we also determine the guaranteed probability of specific values of orbital velo-
citieso Such a prediction thus gives not only the mean orbital velocity, but all
its ot3~er values in this stage of wave development with an indication of their
probability.
BIBLIOGRAPHY
- 1. Venttsel', Ye. S., TEORIYA VEROYATNOSTEY (Theory of Probabilities), Moscow,
Fizmatgiz, 1962, 561 pages.
2. Davidan, I. N., Lopatukhin, L. I. and Rozhkov, V. A., VETROVUYE VOINENIYE KAK
VEROYATNOSTNYX GIDRODINAMICHESKIY PROTSESS (Wind Waves as a Stochastic Hydro-
dynamic Process), Leningrad, Gidrometeoizdat, 1978, 285 pages.
3. Korneva, L. A. and Liverdi, V. P., "Distribution of tidave Elements and Wave
Steepness in Black Sea," KOt~LEKSNYYE ISSLEDOVANIYA CHERNOGO MORYA (MEZHDU-
- VEDOMSTVENNAYA PROGRAMMA SKOICh) (Multisided Ir.~vestigations of the Black Sea
(Interdepartmental SKOICh Program)), Sevastopol', Izd. MGI AN Ukrainskoy SSR,
pp 59-70, 1979.
4. Korneva, L. A., "Statistical Characteristics of Variability of Wave Elements
in the Coastal Zone of a Deep Sea (Results of Registry of Ldaves by the A. A.
Ivanov Wave Recorder)," TRUDY MGI (Transactions of the Marine Hydrophysical
Institute), Vol 23, i4oscow, pp 44-65, 1961.
5. Korneva, L. A. and Liverdi, V. P., "Statistics of Distributions and Spectra of
Internal Waves," Sevastopol', Izd. MGI AN Ukrainskoy SSR, pp 109-118.
6. Shuleykin, V. V. and Korneva, L. A., "Development of a Method for Precomputing
Waves of a Fixed Guaranteed Probability Using the Results of an Investigation
of Waves in a Storm Basin in the Ocean, ISSLEDOVANIYA V OBLASTI FIZIKI OKEANA
(Investigations of Oceanic Physics), Sevastopol', Izd. t~GI AN Ukrainskoy SSR,
pp 39-48, 1969.
7. Korneva, L. A., "Maximum and Energy Width of Wind Wave Spectrum," IZV. AN
SSSR: FAO (News of the USSR Academy of Sciences: Physics of the Atmosphere
and Ocean), Vol 13, No 7, pp 75a-765, 1977.
50
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EFFECT OF GEOMETRY OF PRESSURE DISTURBANCES ON CHARACTERISTICS OF CAP~LLARY-
= GRAVITATIONAL WAVES
Sevastopol' TEORETICHESKIYE I EKSPERIMENTAL'NYYE ISSLEDOVANIYA POVERKHNOSTNYKH I
VNUTRENNIKH VOLN in Russian 1980 (manuscript received 19 Jun 80) pp 55-64 ~
[Article by V. V. Trepachev]
[Text] Abstract: The effect of the geometry of
pressure disturbances on the amplitude
of generated capillary-gravitational
~raves at the surface of a ho~ogeneous
, ideal inc~mpressible fluid of infinite
depth is investigated. The pressure dis-
- turbance is modeled by a nor~al stress
cteanging harmonically with time. The
article examines the case of generation
' of surface wavea by a platform of ice
of finite width oscillating under the
influence of a pressure disturbance at
lts surface. In this problem the pressure
of the platform on the fluid is an un-
lcnown function and is determined by solu-
tion of a mixed problem: '
Capillary-gravitational waves causeci by surface normal pressures have been studied
by the authors of [1, 3, 15]. Problems relating to forced oscillations and diffrac-
tion in the presence of a plate of finite width without allowance for surface ten-
sion were investigated in [1, 2, 14]. The influence of surface tension and the
finiteness of depth was examined in [9].
1. T~e will examine the problem of forced harmonic oscillations of an ideal incom-
pressible fluid of infinite de;~th
a~ ~ aP dv 1 dP
s_
dt P ax a t ~ d r
dcc d v (1.1)
dx + aY ~ D, P= PZ + p 9''
_ -p+~y~- d~ _ P~` r-0,
d x2 ' dt
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P+~'sPw~x~ e LGIt~ ~~y=~D~Q~ tc~,
Cxt 12~~~j-~_��~
,
C; P~ ' +
Here u, v are the components of horizontal and vertical velocity; ~(x,t) is the
form of the fluid surface; o(is the surface tenaion coefficient; g is the acceler-
ation of free falling; 0 is the Rayleigh dissipation coefficient [1].
The origin of the Cartesian coordinate system was selected at the undisturbed f~u-
id surface; the y-axis was directed vertically upward. The movement of the fluid
was caused by harmonic oscillations of normal pressure P* with the frequency W.
The motion of the fluid at the initial moment in time is eddy-free and therefore
even in the presence of Rayleigh forces the Lagrangian tHeorem of conservation of
potential motion remains correct [lJ. Introducing potential using the foi~mulas
~=~ei~dt ~6O~y eit?E ~ ~~?~X~ei~t~
d x ' dr ~1.2~
from (1.1) we have a formulation of the problem for potential
a2? + a12 _ o, p"cx> aU_ ~~U ~~.3>
, y=D.
aX oy r~ Cay aX ay
Q ,~9
w7 (x) i ~y , r-0 . J= (~v 2_~~ ,~1/9 ~ (~.4>
` ~
1
C
~y , a~ , a ~ , z 1 o , ~x =t y~ ~ ~i? ~ .
dX ~Y dx dy ~
Applying to (1.3) the Fourier transform for the x-coordinate, using the contraction
theorem, we obtain integral representations for potential cP and rise of the fluid
surface ~ - - - - - -
y(x,r) ir~ p~~aC)doC eky e-~k(~-,4 e~k(x-d) dk (1.5)
~ 2.~ 9 J J a L ~ ,
p o
~~x~ P� ~~ard f d f e-ik(x-dl+e ~k(x-~) Jd~~ (1.6)
P9 0 -
. d~ k~/+=~k2)-J. (7..7>
P9
Here ~ was determined by formula (1.4); k is the wave number.
The ~ value on the real k-axis with 0 does not become equal to zero; there
are nu free waves in the solution. The use of the complex frequency method W=~1
- i~J 2, ~ 2> 0[6] in this problem gives a result equivalent to the Rayleigh meth-
od. These two methods and others, as well as their correspondence to radiation
~
. 52
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~~ri~u�I~~1~~:+, w~~r~~ Luvc~t:iK~led .Ln (1., 5-8].
2. It has been demonstrated that the dispersion relationship (1.7)
~ ~It ~ ~2~ -J ~ 0 (z.l>
,P9
has three complex roots with 0:
ko , Re~~,~~0, J~ t~o) ~4 ; r2.2)
2 2 ~ Z
~n = - + i y Ko , ~ _ ~p ~ y ? (2. 3)
' Z d
h� _i 9Af-~A'Z, RP .n2>0. (2.4)
- z l ~ y a
tJitti a= o the root k~ _~2/g [1]; computation of the complex root k~ (situat-
ed in the fourth quadrant of the complex wave number k) is convenient by the suc-
cessive approximations metho 13
0 1- F� i/m ~ ~
' _ ' ~nt~ lr11 ~ A ~ ' ~~1,1,J...,
~a-~ ~ /tdhR~P'~ (2.5)
- e-`,~ 2 ~ ~o , ~o ~ ~rz " +d ~ 1/p9) ~ ~ --o .
For the contraction operator A(k) we obtain ttte evaluation
I ~ ~ ~ - ~ C'a~ I ~ I ~a+i - ~a I e ~ , (2. 6)
h~+~ E{ I< eP9Z 2~ ~Pe~r~~7rnk ~>D~, (2.7)
2d fc ~ ~ ~
Here E is equal to a unit length in the selected system of units. With 0 in a
volume of fluid under the free surface an arbitrariness in finding of the solu-
tions appears which is determined by functions of the type
= Le -~~n, z, Ge `~2Z ; (2.8>
_ y = 1Ie-~~tiZ t ,71p `~~z ; (2.`~)
z- x+ i r, ~c y 0, ~0. (2.10)
Here
/n ~ _ - /{'o / 1 t l /1Z ~ /Ilj ' /7L~ ~ l~'~ , /11 ~ ,
(2.11)
z= x- i.r ,,u=D, p"(~)~0, x~ 0.
Ttie solutions (2.8), (2.9) satisfy the Laplace equation and the free-surface condi.-
tion (1.3), (2.11). The solutions (2.8) are exp~nentially increasing in the fluid
volume under the free surface (2.10). The first solution (2.9) corresponds to the
c~ave process propagating from the depth of the fluid; the second solution (2.9)
determines the wave process propagating into the depth of the fluid. For the
53
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solution of (2.9) in the volume of fluid beneath the free surface (2.10) there
are two qualitatively different zones
y-2atx / ko � (2.13)
~ In (2.12) the solutions of (2.9) increase exponentially; in (2.13) they decrease
exponentially. In order to satisfy the condition of decrease of the solution in
the entirP region under the free surface of the fluid (2.10) it must be assumed
that C= D= 0. The arbitrariness in solution of the problem, supplied with the
root of the dispersion relationship k~, with ~l= 0, was studied in [1, 2]. With
~d,~ 0 the solution, determined by the root k, represents a wave penetrating into
the depth of the fluid only in the region x> S,~y < 0, y/x ~ Re(kp)/Jm(kp). The non-
dependence of the Rayleigh dissipation coefficient � on wave scale makes difficult
the applicability of the model in [6]. We will examine the dispersion relationship
for a viscous incompressible fluid [15] (with a Laplace transform parameter
s = i c,~) - "
P
9~-yk'~E1p+~~+1Ek=~j~9A~-~i/1+4~'IimEP ~"0~ (2.14)
where P ~ ;i~l/E , ~Pe~o~O.
The influence of viscosity is determined by the value of the last term in equation
(2.14), the factor p/ (p + k) N 1, k-~s0; p/ (p + k)^' 1/2, k->~ ; the mean is p/ (p +
k)^'3/4, k= (k-~0, k-;oo). -
I~~(~ra~s= f ~~s iiZ
.r......~...~ _ . _ � - - -
; ~0,992
G~II t
i ~
� ~ ~
~ ~
. '
i : p, 996
. ,
.
0,07 ~
. }
. '
. ~
~
~l
' ~
OO~i ! f ~u xz
/,6 Z %y y 8
.
Fig. 1.
iJsing the mean value of the factor p/(p + k) for the wave number and making one
- iteration in (2.14) (with an initial value k= u~2/9), we have an evaluation of
the order of magnitude of
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4~- .
~ ~ 9 ~ ~2.15>
. Here ~ is the kinematic viscosity coefficient. The results of computations of the
k~ of the dispersion relationship (2.1) are reflected in Fig. 1.
3. We will study the form of the fluid surface ~(x) for pressure disturbances
p*(x) exerting a direct effect on the fluid surface. We will examine pressure in
the form - - _ _ _
p~" (x ) = Q 6(x ) , p,*` = 4 . ~ ~ .
. (3.1)
_ Q/Za, ~xia,
.Q Q~-~?, IX~.I~ Q~ '~.~~y~.~ ' . .
p .7fa' ..P~ ',,q. J ~'~a~ .
(3. 3)
p ~ 1 x I > a., . � ~'Q . .
~ -IkIQ ~
pf (X) - ~Qa 2 ~ . py" ~.~Qe . . . .
(3.4)
. ,~~Q t. x ~ . . . . . .
Here
~pz ~ ~ R~ (X ~�e c~txd~ ~ .~y. ~ ~ ~,,,;,4~.
, . �
~ is the Dirac delta function; J1 is a Bessel function of the first kind; Q is the
magnitude of the total force acting on the loaded water surface, related to a unit
width. Using contour integration [10] from (1.6), (3.1~-(3.4) u~e determine
2~x P'~ (K~ c~.~_ i?:' ~r P,~ .f i~~te-~~x! ~~dv~ ~ . . (3. 5)
I , t 2 � ' 1: �
p~ . i+~,~e~P~ . ~P9~~ o ~ ~ , *~y :-.~rv~ig)~ . ~
IX I ~ 0, . (XI a,d., Z;,J; . ..~�::::T: ~
y Q e' a ik e~"~� ~.Q ~ ~ ~ ~ ~ : : -~Y: .
~A\ ! � ~ ~ ~y~'~~~. .
{ ~ � x �
. . . g ~ ..~+~d~; ~ P9 . ~rpq.~. o ~ . . ~ . . . � .
. d ; .
~ [Jcosva+ v;r~in.va ~~1...:.~-:v ~]''d~k',`~:~'~x~'~~-a.fl.. . (3.6)
_ ~1 + v? (1 - d:Y jPy)2~ . ~ : . . . .
~pplying the method of integration by parts to (3.5) and (3.6) we obtain asymp-
eotlc formulas of the fluid surface type
Q ~~~e-i~,ixi aQ ~ ~ ; ' � . .
?i~x~ { r - s t 2 Z . (3.7)
aP9 J+Jd~Y, g . ~L .x ~ . ~ x~ ~ } . ~
/P . . .
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_ _ Q is~n~t'.cr e-~~'~~x~ ---JQ: - I- - Q~~1-- - : - ~ : . ~
?I~X~ ti + - . XjJz (1.- x--z-~.~ , ~ ~ ~ .
a ~ 1 Jd~', g TI l
p9 /P
, . .
~(X~.., Q i1, ~~'.a~?
i~,~X~ _ ~a i z ~ a~~ ~ ~ . � ~ .
~ Q ~ ~ + ~~~o ~ X ~ [ ~ . x ~ 1.}, 3.
P9 /P9
?~x~ ~ Q ik aP-~e ~k,ixi _ JQ ~+.~~2'Ja-JZ-a2-?~ ,
y { ~+JdA~2 xZJ~ l x1JZ
apQ ,~py .
~ x ~ ~ X Q-.-
(3.s>
Here ?~n(x) (n = 1-4) correspond in numb~r to the stresses (3.1)-(3.4); k0 is the ~
root of the dispersion relationship (2.2). Multiplying (3.7) by eiwt, we draw the
conclusion that the solution consists of a travelling wave and level fluctuations.
The level fluctuations decrease exponentially with distance. The level fluctua-
tions for the pressure p3(x) in t*e region of parameters (3.8) are half as great
as for the pressures pi(x), p2, p4�
The principal terms of the asymptotic forms of level fluctuations for the pressures
pi, p2, p* coincided and for the pressures p~`, p*3, p4 have a single type the
type of t~e main ter~ of the asymptotic form of Ievel fluctuation caused by a
deltalike pressure pl(x) = Q ~(x).
In the considered examples the influence of the geometry of the pressure disturb-
ance on fluctuations of fluid level is manifested in the higher terms of the ap-
proximation of asymptotic forms over great distances (3.7), (3.8).
According to (3.7), (2.2), the wave terms decrease with distance exponentially,,
whereas fluctuations of fluid level decrease in conformity to a power law, and at
_ quite great distances from the pressure epicenter only fluctuations of the fluid
level are observed.
The general form of the wave term is represented by the term outside the integral
in formula (3.5). Computations show that an increase in the frequency of the forc-
ed oscillations cJ o~ are fixed, 3 a14 ~/y 2(2.5) ) decreases the wavelength
2 7t'/Re (k~) and the value of the amplitude factor exp (Jm (kp) ~ x Jm k0 < 0,
~ x const. We will study the behavior of wave amplitude without taking into ac-
count the spatial variability factor exp.(-i k~ ~x For pressures (3.1)-(3.4) it
has the form
56 ~
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. . . A ~ : ~r. , A , .s~a k, u ,
. . . . ~ l+Jd,~;
/.P9 2 ~d,r,Z~py
� � . � A~ ' ~7 a~) ~ _ ~o B-k.a . (3.9)
: . ~ � '~a ~ tJo~/Ye / J Q ~ 4 ~a~/ro / J_
J
The values of the amplitudes An (n = 1-4) with each fixed surface tension value
Oc, according to the "compressed images" principle (2.6), in the region of the
complex wave number (2.7), can be investigated as a function of one complex vari-
able kp. The poles of the amplitudes An (n = 1-4) are purely imaginary k~ = t~py/3~
and do not belong to the region of determination (2.7). In accordance with the prop-
erties of the harmonic functions [10] the Re A~, Jm An values cannot attain their
extremal values at the internal points of the de~ermination region and in the
neighborhood of the zeroes of their first derivatives have a saddlelike form. A
special case of internal points in the region (2.7) is real k~ values (~.t = 0,
ideal fluid without dissipation). Hence we draw the canclusion that
I AR(k )~~I,q2 (Re~fo~ I; p,-ksRek~D, a-l=4. (3.~0)
dre
According to (3.10), the moduli of the amplitudes An, with allowance for dissipa-
tion, are greater than the moduli of the extremal values An for a fluid without
allowance for dissipation with equal wave lengths. Without allowance for dissipa-
- tion we derived the formulas
~ 0 or r=-r2 and r=-rl, r=-r2, if x< 0. The conditions for existence
of the stationary points for M1(r) are the same as for M2(r) and the stationary
points for M2 differ from the points for M1 only in sign.
_ Here ul = lim 'C'(r) = y H; u2 =~'(r~); r~ is a positive root of the equation
r 0
~"(r) = 0, the stationary points r=�r2 are governed by the elastic forces of
t}ie ice cover.
Thus, after use of the stationary phases method we find
G2
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~ t p( X 1, ~x~>~~t, _
2 ~I I /
~ _ ~I't' o~ ' ~2~~~x1�~t�
~ . - ~ -,s:::. .
' , . , - (1.6)
: 0.~ .1..1 i I ~ I.'~, ~2.t~:.. . ~ . .
. ~ ~x"~ J.~ . . . i,~.~ . . j:..~ ~L_
+s ~ /q (r LOS _ ~ ~..X t'T C. - ( ~F, k � ~ /~a Z~
~ Ixl . ~ . .....,
where J~ is the rise in the level of the free surface; u, v are the components of
horizontal velocity; 2~ is the Coriolis parameter; h is basin depth (hl in the
- open ocean, h2 in the shore zone).
The rise of the free surface for a tusunami wave running in at an angle from the
open ocean has the form
~ = Ae ~t~r~ ~zy- a~t ~ ~ - ~2~
where A is the amplitude of this wave; m, n are the components of the wave vector
on the x, y axes; b is the frequency of this wave. Using the periodicity condi-
_ tions and seeking a solution of system (1) in the form
{ c~, v, ~ } _ v, ~ } � exp ~ i ~ay - 6t ) } � ~ ( 3)
~
for determining the rise of the free surface in a region of variable depth ~ 2 and
in the open ocean ~1 the ordinary diff erential equations are
9~,~; ? C~2'4~'t-9~,z1~,~,-o, . . c4)
hI J~1 + kZ;i _ l,? f 2~ + h2 R Z-, ~ 2~J s 0.
9 ~5>
Solving equation (4), we obtain
l =Aei~rx+B~e-i~x.~~ G ~4r~ _~2 ,
~ i 9i
where A1 and B1 are arbitrary constants. The general solution of equation (5) is
written in the form
~7 ~ ~f ~~X~ ~ BI ~~x~ ' .
where A2, B are arbitrary constants; ~ and ~ are two fundamental solutions of
- equation (5~, determined numerically by the Runge-Kutta method.
For the rise in the free surface in the open ocean j~l and in the coastal zone,~2
we obtain the expressions
i (inx+ Z r - Qt - i (-n�xtay- Qt ) . ~ .
~ ~ ~ A~ e ~ + ~ B~ e ,
~o
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;1 s rA ~(x) + BI S~(X)1 e ~lar--6~) -
~ ~
- Here A1 is the amplitude of the wave arriving from the abyssal region (we will
consider it to be fixed); B1 is the amplitude of the ~rave reflected in the abys-
sal region; the expression 3n brackets is the amplitude of a wave (dependent on
x) moving along the shore in the coastal zone.
In order to find B1, A2, B2 we will use the continuity conditions for the free sur-
face profile and the horizontal velocities at the boundary of regions of constant
and variable depth, and also the condition at the vertical wall. These conditions
- maka it possible to obtain a system of three algebraic equations whose solution
gives the sought-for values.
For Burevestnik village the proposed method was used in computing the rise of the
level surface with approach of the tsunami wave to the shore. The basin model re-
flects its longitudinal section perpendicular to the shore. The depth of the open
ocean is hl = 4 km. The zone of variable depth is broken down into regions whose
depth and extent in each case with approach to the shore are h2 = 2, 1, 0.5, 0.4,
0.005 km; L1 = 60.5, 16.5, 242, 99, 143 km.
The computations wrre made for waves whose amplitude in the open ocean was 1 m,
the periods being 107, 40, 33 miniites. The choice of periods is explained by the
observational data. It is known [1, 4] that with passage of a tsunami precisely
these periods predominate in the spectrum of ocean level fluctuations.
Source [4] gives the spectra of ocean level fluctuations with the passage of the
close and distant tsunamis of 28 March-January 4, 1964, 14-16 May, 1966, 16-18 May
1968, 11-13 March, 1969, 22-25 September, 1969. Here it is also indicated that
with the passage of distant tsunamis a long-period component appears in the spec-
trum which is absent in the spectrum of variations caused by near tsunamis.
The computations indicated that with the arrival of a tsunami wave from the open
ocean at an angle to the shore, with an amplitude of 1 m and a period of 107 (dis-
tant tsunami), 40, 33 minutes (near tsunami) a wave is propagated along the shore
whose amplitude is greater (by 60% for a distant tsunami and by 70%~for a near tsu-
r_amil than the amplitude of the arriving wave. Since the wave is propagated along
the shore, it can be regarded as an edge wave whose amplitude is 1/2-1/4 of the
- total amplitude.
In [6] a study was made of the record of level variations during the passage of
waves during the Iturup tsunami (1958, 1963). It was demonstrated that the varia-
tions include a special component whose transformation in the transformation pro-
_ cess is characteristic for edge waves on the shelf. The amplitude of these waves
is 0.5-0.25 of the total amplitude and the maximum value is ob3erved in those
cases when the total group of waves approaches at an angle to the shelf. Thereby
a wave component is excited which moves away from the shore along the normal with
an amplitude up to 1/3 of the magnitude of the edge shelf waves.
Taking into account the conclusions in [6], we will assume that the maximum ampli-
tude with approach of a tsunami to the shore, having an amplitude in the ocean of
1 m, will be 7 m(for a near tsunami) and 6 m(for a distant tsunami). A wave will
71
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be reflected from the shore whose amplitude is 0.6 m(for a close tsunami) and
0.5 m(for a distant tsunami).
By comparing in situ observations of hEight of the wave (0.103 m) registered by the
tide gage at Burevestnik village on 12 August 1969 with the passage of a tsunami
(earthquake magnitude 8.2) j5] with the computed height (5 m),with the arrival of
a wave with the amplitude 1 m from the abyssal region, for the actual tsunami we
- obtain an amplitude of the arriving wave or the wave height of the wave at the
tsunami source of 0.2 m. Thus, the amplitude of the tsunami wave with approach
to the shore increases by a factor of 6.
For other actual tsunamis [5] (28 March 1964, 15 May 1966, 16 May 1968, 23 November
1969) there was registry of amplitudes of variations several times less (0.42,
0.15, 0.2, 0.45 m respectively) than during the tsunami event of 12 August 19G9;
accordingly, the height of the tsunami at the source for them will be considerably
less (0.02-0.1 m). .
BIBLIOGRAPHY
- 1. Babiy, M. V. and Yaroshenya, R. A., "Results of Computations of Periods of Nat-
ural Level Fluctuations of Bays in the Example of Kasatka Bay," IZUCHENIYE
TSUNAMI V OTKRYTOM OKEANE (~tudy of Tsunamis in the Open Ocean), Moscow,
_ "Nauka," pp 43-47, 1977.
2. Ikonnikova, L. N. and Yaroshenya, R. A., TABLITSY VYSOT VOLN TSUNAMI (Tables
of Heights of Tsunami Waves), Vladivostok, 1978, 150 pages.
3. Solov'yev, S. L. and Mileteyev, A. N., "Appearance of the Niigata Tsunami on
the Shores of the USSR and Some Data on the Source of Waves," OKEANOLOGIYA
(Oceanology), Vol 7, No 1, pp 104-116, 1967.
4. Yaroshenya, R. A., "Investigation of Natural Level Variations in Bays of the
Kurile-Kamchatka Coast, TEORETICHESKIYE I EKSPERIMENTAL~NYYE ISSLEDOVANIYA
PO PROBLEME TSUNAMI (Theoretical and Experimental Investigations on the Tsu-
nami Problem), Moscow, "Nauka," pp 153-164, 1977.
S. Yarosher~ya, R. A., "Clarification of the Principal Periods of Natural Level
Variations in Bays of the Kurile-Kamchatka Coast, Archives of the Far Eastern
Scientific Research ~iydrometeorological Institute,' Vladivostok, 1973, 403
pages.
6. Aida, I., "On the Edge [Javes of the Iturup Tsunami," BULL. EARTHQUAKE RES.
INST., Univ. Tokyo, 47, 467, No 1, pp 43-54, 1969.
7. Takahasi, R. and Hatori, Y., "A Model Experiment on the Tsunami Generation
From Bottom Deformation.Area of Elliptic Shape," BULL. EARTHQUAKE RES. INST.,
Tokyo, 40, No 4, pp 25-63, 1962.
8. Watanable, H., "Studies of the Tsunami on the Sanriku Coast of the Northeast-
ern Honshu in Japan," GEOPHYS. MAG., No 1, pp 102-115, 1964.
72
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GENERATION OF SEISMIC WAVES $Y TSUNAMI WAVES PROPAGATING IN OCEAN WITH
UNEVEN BOTTOM
. Sevastopol' TEORETICHESKIYE I EKSPERIMENTAL~NYYE ISSLEDOVANIYA POVERKHNOSTNYKH I
VNUTRENNIKH VOLN in Russian 1980 (manuscript received 3 Apr 80) pp 79-89
[Article by I. V. Lavrenov]
[Text] This article is devoted to the problem of
shQrt-range forecasting of tsunamis. The mech-
anism of generation of seismic waves in the
earth's crust by tsunamis is examined. The
resonance conditions for generation are re-
vealed in an examination of the interaction
of tsunami waves with irregularities on the
ocean floor. The spectrum of Rayleigh waves
is constructed in a case when tsunamis are
_ propagated in an ocean ~ahose bottom relief
changes in one direction. Ttie radiation of
seismic waves has a directional character.
The problem of seismic waves generated by tsunamis has a direct relationship to
the problem of short-range forecasting of tsunamis. This problem was raised for
the first time in 1956 by L. N. Sretenslciy in [1]. Then it was examined by S. S.
Voit in [8]. In these studies it was assumed that~tsunamis during movement in
an ocean with an elastic plane bottom excite seismic waves which are propagated
considerably more rapidly than tsunamis. On the basis of observation of seismic
waves it would be possible to ~udge the presence and character of tsunamis,
thereby obtaining preliminary information for some time prior to the arrival of
tsunami waves on the shore.
In source [2] a study was made of seismic waves generated by a tsunami, taking
into account an analysis of elastic displacements of the earth's crust caused
by tsunamis propagating at an angle to the shoreline.
- In this article we examine the possible mechanism which would describe the gen-
eration of seismic waves by tsunami waves propagating in an ocean with an uneven
- bottom.
1. Assume that in a rectangular coordinate system (x, y, z) an elastic medium with
z?-h(x,y,z) is stipulated which is covered by a layer of homogeneous fluid -[h~
+ r!(x,y,t)~ z
Here k is the horizontal wave vector k= ~ k', k" d 6i, d~, dVn, dUn,
are the components of Fourier-Stieltjes stresses, potentials and displacements.
The initial functions d~ and dVn are obtained in the form of 'Mellin integrals as
a result of substitution of (2.6) and (2.7) into (2.4) and (2.5)
x.i.. -Z~d pt -
z, t~'t~ J~~'~'P~ e ~ dp ?
arr�
_ X~~M (2.9)
_ d~~~'~~ Z~ tJ=1 J. ~~'~?P~ t-z,~~e ~ Pe~P ~
X-[
where -
~~I+ a~~ ; ~I+ -~t~
We will take the residues at the poles of the integrands (2.9) with integration in
the plane of the complex variable "p." As a result we obtain expressions determin-
ing the components of displacement of the elastic medium in a Rayleigh wave
L
d!~� t, t~=~ ~i~ z~ S s1a w~t-r~ d~~ f~l v, (2.io>
o
where vR is the ve~locity of the Rayleigh wave, equal to 0.9194/6 (on the assumption
~hatf-~ Cin(k,z) is a tensor, setting in agreement the components of the vector
U and the components of the vector 6.
~ 76
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� We obtain the correlation between the energy characteristics of the random field
- of the limiting stresses and the corresponding characteristics of the field of the
Rayleigh wave. In the space of wave vectors Tc' we introduce the spectrum of the
field of stresses with the time shift t
~l t ,~6dd~ ~t , d~4~~Fo+~,~+t~ (2.11)
~~C,i, ) d, .
Here the brackets denote theoretical-probabilistic averaging, the asterisk denotes
a complexl.y conjugate parameter. With i= j the components of the tensor Di~ repre-
sent spectral functions of each stress component. With i~ j the components of
Di~ are the cross-spectral functions of the stress components, which, spea~Cing in
general, are ~omplex. If the stress comp~nents are~stationarily related, tYee compon-

- ents of the D1~ tensor are not dependent on the initial time t0.
We introduce the spectrum of components of displacements in the Rayleigh wave
'Sirni t z =
B~~o~(drd)�-c~L , 8~~~/!z~-~, ,3
61~=_~e_cz~z~, B =d(d}d~t GZ~y1 8~=~9[~n2-~ (d+~)],
11
6~~S~H(e~--~(d+d~] , 631=-~H[~~r+n.(d+~d)],~=(d+a'~+~Z,
P= e1+~2k?, d= r+AfcT, ~f- naz+zz, c= 9H. s~0.
The integrals (1.4) are computed using the theory of residues; the poles ~ are the
roots of the cuhic equation
d'~+ 2cld 1+ ~d1~~o2~d + dc1k2~0, (~.5)
which are easily found using the Cardano formula [10]. We will examine axisymmetric
waves. Then the double integrals (1.3) can be transformed into single integrals. We
wi~l not use up, v~, u, v, but u0~, v0*, u~, v*, where u~*, u* are the radial and
v~ , v~ are the tangential components of the velocity vector. Then transforming to
polar coordinates both in the integxals (1.3) and in the transforms u~, v0~ Z 0~
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and also ~arrying out a transition from up, v0, ~p and u, v, ~ to u~, V~* and
and u*, v,~ respectively, we obtain
- .e _
~ ldi~ ~o* + d
Z o" + d
~s 0 1 '1~/f/P ~k s ~/{R' t~ '
~ (1.6)
v"s ~ . ~dL ~c; t du v" + d~ ~o ~ ~,~~R~~= ~ v~ t) ,
S .
o
~ J ~d~~~~ +d? ~~~o~Jo ~~R)dlr= ~ t~ ~R~t~- (~.s)
a 0 ~s~
Here ~l0' ~1 are Bessel functions of the first kind;
Q= ~~�~~3~3'~4~1)Z~ P,�-d1/~*p2; qa-Zd'~~17td~c?~f2 ZB')~~;
~
~ f ct,'~,P ~J~ ~~f~ d~P ~ o"CR~ RJ ~if,Q~dR; Ia= f~,~~P~RJ,~kR~d~i�
0 0 ~ .
The coefficients d.. (i, j= 1,3) entering into (1.6)-(1.8) have a different form
in dependence on t~i~ sign on Q:
a) Q< 0,
~ - (1. 9 )
di~ ~ di ~d~ + d) s~ , d
Z- e/( ~ eli si ? di3 � 9 k~~�ti + d~ S~ ~
~~r t / i~?
J ~
dL - d/Z' ~11,�~~ l�(1(�(~rQ~)tG2/f1,3~, ~9hj ~ Si 1
1
d~~-y,~1 ~ (~`+d~s~ . d~1=-H~~'1 ~ s~ � d,1~~'~ ~ ~~d~'d ~Z+e1 ~s~'
c i ~ u/
s~ =eXP ~d~ t~~~'1�(tZt 4dd~ + B~, B- d Z+p? d~ - 2 P~~~ cos ~~P/3~-a/3,
o(~3s-ZV
-~cos ~(y�~)~,~)-a~9, a-?d, cosc~+-4~~`l-P~~~~~~ ; .
b) c2> o.
Q;,- ~ a, t d) a,-z ~,u2 a~ i drr'~if [ a~ ~i+ z(~u~ ~rc3+~�~~~
~?;3~9k{(cz~+d~~1 t1 ~(~t,~Q~~src~~�~C1d~+~u)J~]} , dL=-~1.
`
d2-~Y{ f Q, c~+a,~~2C2) ~*1 [~Zt ~~'ZCI/~J-a9CJ?-Zd ,~2C1/ 1�
J
d~~-9ei~z [a~Z (~,~~t,taJo,~~)] ,
cl~~--Hk1~a~+d~a+z~~f1i dd~F~rta,~Zd~+~~J~}, d~1=d~ H~y,
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d~=,r! (~a~td~+e1)~,.z[~i+6da1f-d~C,t1d~B~t+ydf)+
l
'�e2~d2 J ~~-La3~B~aZtd)-df-~,~e +d1~~~~~}' .
~
(~.~o)
Q~=ll,~y1-~~9, a2=-~A~''~'z~~2-a~~, a3~~~A-A2~ 2~
AZ= ~ 9~2t~'~;~[=a1 ,u=~f-B, J=3~a? -a~}+~4da2+6
~i = f~9a1+ 4d~+ Ba1, ~~~a1 + Zd, ~I=exp (a,t ~~~v'a~ t yda~+6~,
~CZ =,~f2+ ~da1 f t B~a1- ct~ d~c~, J2 =~idf+28cz2 -,ud,
~1~=cos (ta~) , ~=si~ ~ta~~ ,
z= lexp(a1t~/~f(yf+6dV~~+6 ~6(�Z -a~)+BdaZ+B)~.
- If it is assumed that the friction and horizontal viscosity coefficients are equal
to zero, then (1.6)-(1.8) will provide a solution of the similar problem for an
ideal fluid [8].
A further investigation of u*, v~, ~ was made numerical~~ using formulas (1.6)-
(1.8) for the following model laws of change in u~*, vQ and
_ a) ~o ~ Bo Zy~e ~Rexp~f?'~Z~~ (1.11)
b~ ~a = Po Z f?e RexP~ y~R 2) ;
~o =a, ez~~~R~~~
For the figures represented below ap = 1 m, b=/~0 = 1 m/sec (max u~ = max v~*
= 1 m/sec); H= 4�103 m; 45�; y~= 0.38�1S'9; = 0.23�10-9; the numerical
values of the W and d' parameters correspond to the characteristic half-width Ll
of the initial disturbance, equal tu 105 m. By the half-width of the disturbance
f~(R) is meant the maximum of the possible values R= L1, for which fp(L1) = 0.1
max f0(R). '
R
In [8] an analysis was given of the velocity field and the form of the free sur-
face of an ideal homogeneous rotating fluid caused by initial axisymmetric dis-
turbances. It is shown that they cat~se the for:nation of time-attenuating annular
waves and some stationary barotropic eddy at the center of the disturbance. Below
we give an analysis of the influence~ of horizontal and vertical friction on the
process of generation of these wave:: and time-attenuation of the indic~ted eddy
formation.
Now ~oe will examine the influence of vertical friction on the process of evolution
of the initial rise of the free surface of the �luid and tangential velocities
(r 0, A= 0). Curve 1 in Fig. 1 gives the shape of the generated wave for the
86
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initial rise l~ p in the form (1.11). Ita profile is characterized by a rising wave
and by a dropping wave which follows it. A comparison of curves 1, 2 in Fig. 1 re-
veals that vertical friction does not lead to qualitative changes in the shape of
the wave but its amplitude value can be substantially reduced. This effect, as
might be expected, is intensified with an increase in the vertical friction coef-
ficient r. For example, if the value max ,~(R,t)~at the time t= 30 min with
r= 0 is equal to 0.137 m, then with r= 10-4, 2'10-4, 3�10-4, 10-3 sec'1 it can
decrease to 0.123; 0.115; 0.107; 0.067 m respectively. With an increase in t the
difference in the amplitudes of the l~ + waves with r= 0 and r~= 0 is intensified.
This conclusion is illustrated in the table, where column 1 gives the values
with r= 0 and column 2 gives the values with r= 2�10'4 sec'l.
1
~j M ~
0, I 4
~05
~ J00
900 ~P K~
, -0,05 ~
~
J .
!
~ Fi.g. 1. Profiles of ?~3 wave for t= 30 min: r= 0, A= 0(curve 1); r= 3�IO-4
sec-1 A= 0(curve 2); r= 0, A= 105 m2�sec-1 (curve 3); r= 3�10'4sec-~, Q=
105 m~�sec-1 (curve 4).
Figure 2,a illustrates the influence of vertical friction on the field of the tan-
gential component of the velocity vector v* = v3. The curve marked with the symhol
- ~ describes the v* distribution in a stationary geostrophic eddy arising from
the initial rise of the free surface of an ideal fluid. It can be seen that in a
viscous fluid this eddy attenuates with time. With an increase in the friction co-
efficient r the attenuation of the eddy is intensified and accordingly the life-
time of the eddy is reduced. We note that allowance for vertical friction exerts
no influence on the position of the maximum v3 value.
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~ �/Dy /00 _ _ 200 ~P M
2 ~
-1
5
-4
zo
a
~
-6
3 ~~y ~P ~ra
2
-1
S '
10 �
6
~
10
~o .
Fig. 2. Distribution v�104 (msec'1) for different moments in time (figures over
the curves): a) r= 1033 sec'l, A= 0; b) r= 0, A= 105 m2�sec'1.
It was demonstrated in [8] that any initial distribution of tangential velocity
(v~) in an ideal fluid in the case of long waves virtually does not change in time.
Table
t minutes t Mx 1 2 3 4
10 0,212 0,20b 0~201 0,200
- 20~ 0,188 0,158 0~162 0~146
30 . 0,18? .0,118 0,12? O~11S
40 0,116 0~098 0,109 0,090
5p ~,104 0,080 0~088 0,076
80 ,09& 0,071 0 ~085 0,~87
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The transpiring insignif icant changes in hydrodynamic characteristics are attrib-
utable only to the effect of Coriolis force. Under the influence of vertical fric-
tion, with an increase in t the initial distribution of the tangential component
of the velocity vector is considerably deformed, as can be seen from a comparison
of curves 1, 2(Fig. 3). With an increase in t the amplitude values v'~ = v2 de~
crease, but the position of the maximum value, as in the case of the initial dis-
placement of the fluid surface, to all intents and purposes persists. Numerical
computations made it possible to estimate the lifetime of such eddies. For example,
with r= 3�10-4 sec-1 a decrease in the maximum velocity value by a factor of 10
occurs during a time interval of about 2 hours.
~Z 'y~L m�sec'1
/
/,0
3
0,5
t
4
0
SO /00 R KN
Fig. 3. Distribution v2 (m�sec-1) for t= 0(curve 1) and t= 1 hour: r= 3�10-4
sec'1 A= 0(curve 2); r= 0, A= 105 m2�sec-1 (curve 3); r= 3�10'4 sec-1, A=
105 m~�sec-1 (curve 4).
3. We will examine the influence of horizontal friction on the process of evolu-
tion of the initial rises of the free surface and the field of tangential velo-
cities (r = 0, A~ 0). Horizontal friction, like vertical friction, favors a more
rapid decrease in the amplitudes of the main wave and the descending wave in com-
parison with the case of an ideal fluid. A comparison of curves 1, 3(Fig. 1)
reveals that horizontal friction leads to some "spraying" of the wave in a radial
direction. The dependence of the amplitude of the main wave (m) on time t(min)
with r= 0, A= 5�104 m2�sec-1 is given in column 3 in the table. With an increase
in A the attenuation of with time will evi4ently occur more rapidly.
Figure 2,b shows the process of evolution of the field of tan~ential velocity
caused by an initial rise of the free surface. It can be seen that with time due
to horizontal friction, as well as due to vertical friction, there is a decrease
39
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in the amplitude values v3 and entry into a stationary geostrophic regime of move-
ment does not occur. In contrast to the case r=~ 0, A= 0 under the influence of
horizontal friction the position Rx Iv3(R)~ with time is displaced into the re-
gion of high R values.
Horizontal friction also exerts a substantial influence on the initial distribu-
tion of the tangential component of the velocity vector, as follows from a compar-
ison of curves 1, 3(Fig. 3). It can be seen that the amplitude values v* = v2 de-
crease and the position zn~x jv2(R) ~is displaced in the direction of high R val-
ues. Computations show that with an increase in the A coefficient the field of
tangential velocity attenuates more rapidly and therefore the lifetime of the eddy
decreases. It is several hours. For example, with A= 105 m2�sec-1 for t= 2 hours
/?L1lX ( yt \R ~ a0,,~ i~taz ~ r," I.
R ~
. 4. ~de will examine the joint effect of vertical and horizontal friction (r ~ 0,
A~ 0). An analysis indicated that the joint influence of vertical and horizontal
friction leads to a decrease in the amplitudes of the head wave and decreasing
wave and also to a"spraying" of the first in a horizontal direction. This is in-
dicated by a comparison of curves 1 and 4(Fig. 1). Wave attenuation, as might be
expected, with A~= 0 and r~= 0 is moi: significant than when only vertical fric-
tion is considered or only horizontal f~�i;.tion. This is seen clearly dlso from the
table given in section 2, in which column 4 gives the numerical values 1,~+ (m) for
r~ 2�10'4 sec-1, A= 5�104 m2�sec'l. The computations also show that in general
a change of the r coefficient exerts a considerably greater influence on the wave
characteristics than a change in the A coefficient by the same number of times.
The joint i.nfluence of horizontal and vertical friction on the process of evolution
of the initial distribution of the tangential component of the velocity vector v~
is illustrated by curves 1 and 4(Fig. 3). It can be seen that with an increase in
t under the influence of friction the amplitude values v* = v2 decrease and the
position ~x Iv2(R)I is displaced in the direction of large R, as occurred due to
horizontal friction. Hok:ver, this decrease in v2 with one and the same parameters
of the prob~em exceeds the corresponding change occurring with allowance for hori-
zontal friction a.lone.
BI~3LIOGRAPHY
1. Sretenskiy, L. N., TEORIYA VOLNOVYKH DVIZHENIY ZHIDKOSTI (Theory of ~dave Move-
ments of Fluid), Moscow, "Nauka," 1977, 816 pages.
2. Kochin, N. Ye., K TEORII VOLN KOS}~I-PAUSSONA (On the Theory of Cauchy-Poisson
Zdaves), Vol 2, Moscow-Lenin~rad, Izd-vo AN SSSR, pp 137-154, 1949.
3. Voyr, S. S., "Long Waves and Tides," ITOGI NAUY.I (Scientific Results), Vol 2,
- SER. OKEANOLOGII (Oceanological Series), Moscow, VINITI, pp 46-89, 1973.
4. Cherkesov, L. V. , GIDRODINAMIFCA POVERKHN(?STNYI:N I VNUTRENNIf:H VOLN (Hydrody-
namics of Surface and Internal LJaves), Kiev, "Nauka Dumka," 1976, 364 pages.
90.
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5. Potetyunko, E. N., Srubshchik, L. S. and Tsaryuk, L. B., "Use of the Station-
ary Phase Ptethod in Some Studies of the Theory of Wave~ at the Surface of a
Viscous Fluid," PMM (Applied Mar_hematics and l4echanics), Vol ~4, No 1, pp
153-161, 1970.
6. Vol'tsinger, N. Ye. and Pyaskovskiy, R. V., OSNOVNYYE OKEANOGRAFICHESKIYE
ZADACHI TEORII MELKOY VODY (Principal Oceanographic Problems in the Theory
- of Shallow Water), Leningrad, Gidrometeoizdat, 1968, 300 pages.
7. Kagan, B. A., GIDRODINAMICHESKIYE MODELI PRILIVNYKH DVIZHENIY V MORE (Hydro-
dynamic Models of Tidal Movements in the Sea), Leningrad, Gidrometeoizdat,
1968, 220 pages.
~
8. Dotsenko, S. F., Sergeyevskiy, B. Yu. and Cherkesov, L. V., "On ~volution
of Axisymmetric Disturbances in a Fluid," MORSKIYE GIDROFIZICHESKIYE ISSLEDO-
VANIYA (Marine Hydrophysical Investigations), No 1, Sevastopol', pp 15-31,
1978.
9. Dotsenko, S: F., Sergeyevskiy, B. Yu. and Cherkesov, L. V., "Influence of
Friction on Propagation of a Long Wave Caused by Initial Displacement of
a Free Surface," POVERKIiNOSTNYYE I VNUTRENNIYE VOLNY (Surface and Internal
Waves), Sevastopol', Izd-vo AN Ukrainskoy SSR, pp 87-94.
10. Korn, G. and ICorn, T., SPRAVOCIiNIK PO MATEMATIKE DLYA NAUCHNYKH RABOTNIKOV I
INZHENEROV (Handbook on Mathema,tics for Scientific Workers and Engineers),
Moscow, "Nauka," 1974, 832 pages.
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EFF~CT OF VISCOSITY I1Nb BFTA-~FF~CT ON GENERATION OF LONG WAV~S IN OC~AN BY
ATTIOSPIi~RIC WFVES
Sevastopol~ TEORETICH~SKIYE I EKSPERIPIFNTAL'NYYE ISSL~DOVANIYA POVERI:IiNOSTNYKH I
VN~iJTRENNIKH VOLN in Russian 1980 (manuscript received 19 Jun ~30) pp 99-107
[Article by V. F. Ivanov]
[Text] Abstract: Tne author gives the derivation of
an equation for the complex amplitude of a
wave taking into account the influence of
viscosity, the ,g - effect, bottom relief and
wave shearing stresses. Simple formulas are
derived for wave velocities, taking,into ac-
count the surface and tiottom fricrion layers,
as well as formulas for wave shearing stress-
es expressed througti periodic fluctuations
of atmospheric pressure. It has been estab-
lished that the influence of viscosity and the
p-effect is manifested for the most part in
the resonance region and leads to a change
in the resonance amplitudes and frequencies.
It is known that atmospheric disturbances at the ocean surface [1-3] are one of
the important sources of generation of internal waves in thP ocean. At the same
time, in the upper surface layer of the ocean, due to allowance for turbulent
friction, th~re can be inte:~sive turbulent wave movements caused by fluctuations
of atmospheric pressure. In most studies [1, 3, 4] devoted to an investigation of
the houndary layers and generation of internal waves only the bottom boundary
layer is talcen into account and also the boundary layers forming at the discon-
~ tinuities of a multilayer f.luid. At the ocean surface it is customary to stipul-
ate the normal stresses and shearing stresses are usually not taken into account
at all [1-4] or are taken into account but are not related to fluctuations of
atmospheric pressure [5, 8, 12].
In this article we give the derivation of an equation for tlie complex amplitude of
a wave and formulas for wave velociti~s, taking into account the surface and bottom
~ friction Iayers,and investigate the joint influence of viscosity and the ~-effect
on the generation of long waves in the ocean by atmospheric waves.
Within the framework of the linear theory of long waves, with allowance for ver-
tical turbulent viscosity and the effect of Coriolis force, assuming the movements
to be periodic, we will write the initial equations in the form
92
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. A . ~o-i
R
7
r''
l : ~
i
J y .~i
~ ~ 't
; ; v~ �
~ ~ f
. ~
, �
~
' � I
Z
I -
J
09BB q991 0,996 l,000 /OOf~ l,OOB /,O/1
Fig. 1.
Table 1
y e � 0 SO , 180 27C~
t0 425~ 3 398, S 427, 6 458, 8
(1,0028) (I,Q012)., (0,9887) ( 1,0012)
_ 2Q 424,4 97S,8 426,8 480,6
(I,002R~ (1,OO1Z) (0~9998~ ~ (1~0012)
~ 423~ 1 359~ 9 425,4 816,?
( 1 ~ 0035~ ( 1, 0012 ) ( 0 , 6A89 ) ( 1, 0012 ) .
~ 421, T 351 ~ fl 428, 4 530,8
( 1,002~) ( 1~0012) ~ (1,0000) (1~0012)
420, 0 349, 4 421 ~ 8 529~ 0
( 1, 0022 ) ( 1, 0012 ) ( 1 ~ OC02 ) ( 1 ~ OOl2 )
80 418~ 8 9S5~ 1 418,@ 51 3 '
( 1,0020 ) ( 1, 0012 ) ( 1 ~ OOiD4 ) ( 1 ~ 0012~)
- 417, 5 388~ 4 418~ 4 48~~ 5
( 1,0018) ( 1~0012) ( i,Q007) ( 1,0012)
80 416~ 8 889~ T ~ 417~1 449,~
. (1~0015) ( 1~0012) (.1,0010) ( 1~0012~
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2
, a,, a~
~ 6~ ~ ev = po d- . ,
~lj
d2v ,
~6~-e~= 1 -a dZ:
P cz~
" a~ + a~
dz a 0
-i~~+ S Ca~ ay c3>
~
with vertical boundary conditions. At the ocean surface with z=- Zr(x,y)
P~Pa ~ (4)
_ � pa~ a~ _-Z~ , a~ _ _ y , ~5~
aZ az
at the ocean floor with z= H(x,y) (the tiottom is considered fixed) for velocity
of a wave current we apply the attachment conditions
- (1 sV " W ~6)
Here 2cJ sin'~, ~ is latitude; the x-axis is directed to the east, the y-axis
is directed to the north and the.z-axis is directed downward. Since henceforth we
- will be interested in wave movements in the ocean generated by atmospheric waves,
pa and p represent wave pressures in the atmosphere and ocean, tX and Z are wave
shearing stres~ss, ? is the coefficient of turliulent friction of water, 102�cm2/
sec [7].
For a homoger_eous ocean, witli (4) t~iken into account, ~ae have
P =~�a ~ P�91
And the system (1) and (2) is reduced to one equation relative to u
. _
~1 ~y
y+ 2 i 0~ dz -~61-11~~ L2~P) ~
d z a.~
where
LR _ ~ t ( a ~ .
1p. [0't(-~1^`e] Ox dy
The solution of (8) is sought in the form
_ ~ i r Cn ' y~ i~ r/) R C1 . ( 9)
~ A~/ `
/i.?
where _ L ~p`
~ p e ) �~n1 t ~R B~ (/-~).~zl ~ 1 ~
- C z ~
z
Q,- f-1 ~'e . 6~ l.
�!n 'r 1 J
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Taking (5) and (6) into account, for Cn ra.e ohtain
C ~ ~ l~i~t* t,~~(/-~k~K? L~'o~~ C~z _G `~d"'(.y-t) (10)
n y~ K 1 x ~ r s .y'~
In (10) the first terms correspond to the drift velocity of the wave current; the
second terms correspond *_o the gradient current and the third terms correspond to
the bottom current.
In the derivation of formulas (9) we will neglect the terms
e- 2 (1-i ) arH
in comparison with 1, since everywhere we assume ttiat the depth of the ocean is
greater than the thickness of the lioundary layer. They are correct for 6> e at
some distance f rom the critical latitudes.
We will express the drift part of the wave current through fluctua.tions of atmo-
- spheric pressure. For this purpose we will examine a system of eq uations for per-
iodic movements in the near-water layer of the atmosphere, which on the assumption
of the linear theory of long waves for periodic movements has the same form as (1)
and (2). In this case we replace ~ by Y' is the coefficient of turbulent
f riction of air, v' = 104�cm2�sec [7, 9]); by p is meant atmospheric wave pres-
sure.
As the boundary conditions we take
u= uo, v= vo with z= 0; u, v are limited with z=- (11)
Then the solution of (1) and (2), 1iy analogy with [8J, for the atmosphere is writ-
ten in the form
1 ~
ll=~ Dn , ~s ~ ~ ~ D~ , '
,r�i (1 Z )
where -!)d~ z - i)d;~z. z ~
A G * B e + f F~,sh C~' ~)d~ ~Z-~~
R, /r, R
~ (13)
Here
, 6,. e F= fl dp~_~~n aP ~
1 J' ' y p, d,~ J` d x a'r .
Satisfying the boundary conditions (11) and differentiatinR u and v for z for
p J d cc i d V
( l ,
Z~~ � ,\dZ ~z�a ~ ~y p~ J dt/z�o (141
we obtain the expressions
Zx~ L y~~~(~~)RTz ~
L~ ~ ~ ].5 ~
~o.~
where ~ _ _ (I-i ) ~ i d,od + ~ ~ ~R d~oa , 1 .
R ydh dx ay ~
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In (15) we will neglect terms. with u~ and v~ s.ince thex are small in comparison
with the otfiers, but wa will assume that p= pa with z= 0. These expressions
are an analogue of the formulas for the components of wind s.hearing stress [7].
In (10) we ~rill replace ~g by ~y using formulas (15). ~~ien we have
CR.s.._` r~~ LR /Pa~e Wi)~z_~hC~p~+~s~P~1,~/ e-(i-i)dR~N2) (16)
V v ~
Substituting u and v from into (3), with (7) and (16~ taken into account, for
the complex amplitude of a wave on the ~-plane we ob tain
ce ~ ~2_eZ ; +QxH~'-c~yH. ~L _
ae~~+f~H, t 6 ~~c'~ ~i) * 9 a~' aX 17
I n ~ ~ N ( )
y
_~j+ ~ a;~,~_ ~ 4r dPa ~
where ~ - QI
v
~ Pa ; 1(N,I~~ dH d~,+ aH~ ~
p,q ' ax aX ar ay
- ~~H = aH a~ _ aH a;, . z~e _ ~ ~62+e1) .
` ' dx d dy dz 4X= ~Z ; Q,, " ~ (62_e~ ,
z ~ r t~te)'~~+ (o-~)~~Z 1~ 2~a'cos ~p, a-e~4 ~v~,~.
4r" y 6(62-e ) ~ J
In the derivation of equation (17) we will neglect the secondary effects of the
variaLility of ~ and H, related to viscosity, si~~ce we are considering a deep
ocean ~H> 5�0~ -1) . Their contribution does not exceed several percent in compar-
ison with the ~undamental terms II ana IV, which were retained. Fur a deep ocean
and small periods (about several hours) it is also possible to neglect viscosity
(term V), leaving term I.
Now we will integrate the continuit~ equation
' acc + dv ? aw _ o
ax ay az c~s~
for z from 0 to z, and using the buundary condition w= i61~ at the surface, we
ob tain ~
w i w,~ . (19 )
a.~
where .
w _ ~ a p e-~~-~~d,~z~
a
a 2 ~t B~ 1~ d
_ pe~ ( ) ~
,a z a,o ~ 1
+1eP-~d+~~~K`~ f ~dx J~ dy 1 +
( ~~P} _ i~ j~z J~N~P~ _ (~+t d pl -ci-i)d~, (~-zjl
. t lI 1~ ~ p J.
~
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Specific computations were made for an ocean of a constant depth. Assume that a
plane wave of atmos.pheric pressure moves over the ocean
PQ ~ Po e t~,rr~- ~'t) ~ (20)
where r= x cos e+ y~ sin B; B is- tfie angle between the direction of wave propa-
gation and the x-axis; Pp is tfie amplitude of a wave of atmospheric pressure. In
an ocean of constant depth it generates a wave in the form
~ _ ; e ~t) ~21~
0
and with the amplitude From (17), with pa = p~eikr and ~ ~ikr taken into
account with H= Fi~ = const, we find an expression relating the complex amplitude
of a wave in the ocean to the amplitade of a wave in the atmosphere
_ j."Po j 11(!+i ~ d'2 J~I ~ 4r~o ~ -i -'Jl ' (22)
~ d QrH, t~(Q cnsB+ aQ si~ B~ k, J
r x
where k
H
~ ~o' ~ ~!V9. ~
Here k~ is the wave number (space frequency) corresponding to free oscillations
in the absence of viscosity and tfie ~-effect. Using ;22) we determine the wave am-
plitude (absolute ~0 value) and its phase
y~ R2t Iz , y~ arctq R� (23)
where R and I are the real and imaginary parts of formula (22). If it is assur~-
- ed that = 0, in the absence of the ~-effECt resonance occurs when
~ A i po (I- qr)=+4r , y~ Q~.~~
~P I-~y ' P Qr Qr (24)
~de will assume that H= 1000 m, 30�, p~ = 100 Pa. It follows from (24) that
resonance sets in with jf > 1, that is, with a spice frequency higher than k~ and
a phase close to 90�. Since qT � 1, the shift is small. With ~f = 1 the amplitude
already decreases by a factor of f and the phase is equal to 45�. Here and in the
- text which follows A and A~ are indicated ir, centimeters.
The figure shows the dependence of the dimensionless amplitudes (p0 -~P Og~~~O~
on the dimensionless space fr.equency The dashed curves 1',2', corresponding
to ~eriods of 2 and 3 hours, were computed in the presence of viscosity; the solid
curves 1-4, relating to 3 hours, for e equal to 0, 90, 180 and 270�, were obtain-
ed taking ircto account both viscosity and the,~effect. It can be seen that allow-
ance for vertical turbulent viscosity leads to a shift in the resonance curves to
the right (to the left if the dependence of dimensionless wave amplitude on the
dimensionless time frequency 6/ U ~[2] is examined). This shift increases with
an increase in the period, but the wave amplitude decreases. As a result of the
joint effect of viscosity and the ~-effect, even for identical periods,in depend-
ence on the e angle, there is a change in wave amplitude and a shift of the reson-
ance curves.
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For example, if the wave propagated to the east ( B= 0) the resonance curve 1 is
shifted to the right relative to ttie dashed curve 2', computed in the absence of
the {~-effect. For a c~ave propagating to tfie west (B = 180�) the shift of curve 3
occurs to the left. Their resonance amplitudes are virtually identical. If the
wgve is.propagated to tfie south or north ti~ere is no shift of the resonance
curves but tfie res~nance amplitude with 0= 90� (curve 2) is less, whereas with
g= 270� (curve 4) it is greater than when only viscosity is taken into account.
With an increase in the wave period these differences become greater. This also
corresponds to an increase in the circles, one of which, with T= 3 hours, repre-
senred hy the dashed line, is sliown in this figure. Each point taken on it cor-
- responds to the resonance amplitude value computed with a smooth change in the
B angle from 0 to 360�. An analy.sis or" the results of c~mputations shows that the
inf luence of viscosity, and also the joint influence cf viscosity and the ~-ef-
f ect on wave amplitude,is manifested for the most part in the region close tc
- resonance (0.98 1.02). Beyond it these effects are not significant (all the
curves for different periods and e angles virtually merge). With respect to wave
phase, with ~ it is negative, whereas with ~'y~ it is positive, that is, a
wave generated in the ocean with an increase in ~'first lags in phase behind the
atmospheric wave and then for ~j >~p begins to "outpace" it, cbanging very rapid-
ly at 180� witli t~-~ ~P. We note tfiat all the curves in the figure were co~:puted in
the presence of wave shearing stresses in (22). However, allowance for them does
not lead to a substantial increase in wave amplitude for the considered periods,
Sut i~ more important in computing the velocity of the wave current.
~
~ Now we will clarify how a change in the Coriolis parameter exerts an influence oi~
wave amplitude. It was demonstrated in jl] that for free oscillations, correspond-
ing to small periods of about 1-8 hours, a change in the Coriolis parameter exerts
little influence on the wave amplitude. For forced oscillations such will be th~
case only far from resonance, whereas near it the role of the Coriolis parameter
increases.
The table gives tlie values of the resonance amplitudes An (T = 2 hours, 0) as
a f unction of latitude ~ for four values of the B an;;le,equal to 0~ 90, 180 and
27Q�. The figures in parentheses correspond to ~'P values corresponding to these
resonance amplitudes. tt can be seen that if the tiave is propagate3 to the west or
east, its amplitude varies littl~ with latitude. However, the greater the lati-
tude, the greater will be the ~'value at which resonance is attained for 180�
and the lesser will be the ~ value for 0�. If the wave is propagated to the
so uth or north, resonance is attained with virtually identical ~J values, but with
an increase in latitude the wave amplitude for e= 90� first decreases and in the
region of the middle latitudes attains a minimum value, and then :.ncreases. On the
c~her hand, for an angle B= 270� the rescnance amplitude of the w~~ti~: dttains a
maximum value near tl,e middle latitudes.
Summary
An equation was derived for the complex amplitude o� a wave tak'ng into account
the influence of viscosity, the ~i'-eftect and variable ocean depth. Simple for-
mulas were also derived for the velocity components of a wave current, taking the
siirface and bottom friction layers into account,and for wave shearing stresses
caused hy periodic fluctuations of atmospheric pressure.
ns
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It is shown that the influence of viscosity, the ~-effect and change in the Cor-
iolis parameter on wave amplitude for small periods (of the order of several
hours) is manifested for the most part in�tfie resonance region and leads to both
a c}iange in the resoi~ance a~plitude of the wave and to a shift in the resonance
frequency.
_ BIBLIOGRAPHY
1. Cherkesov, L. V., GIDRODINAMIKA POVERKHNOSTNYKH I VNUTRENNIKH VOLN (Hydrody-
namics of Surface and Internal Waves)~, Kiev, "Naukova Dumka," 1976, 384 pa.ges.
2. Zadorozhnyy, A. I. and Khartiyev, S. M., "Long Internal Waves in a Continuous-
- ly Stratified Viscous Fluid Caused by Atmospheric Pressure Fluctuations,"
POVERKHNOSTNYYE I VNUTRENNIYE VOLNY (Surface and Internal Waves), Sevastopol',
Izd. MGI AN Ukrainskoy SSR, pp 153-162, 1979.
3. Levkov, N. P. and Cherkesov, L. V., "Boundary Layers of Surface and Internal
Long ~Javes.," IZV. AN SSSR: FAO (News of the USSR Academy of Sciences: Physics
of the Atmosphere and Ocean), 3, 9, pp 304-310, 1973>
4. Levkov, N. P., "Diss~ipation of Surfac~ ~daves Generated by Periodic Atmospheric .
Disturbances," MORSKIYE GIDROFIZICHESKIYE ISSLEDOVANIYA (Marine ~iydrophysical
Investigations), No 4, Sevastopol', pp 76-83, 1973.
5. Vapnyar , D. U., PLANETARNY`IE VOLNY I TECHEi1IYA V EI:VATORIAL' NOY ZONE OI:EANA
(Planetary Waves and Currents in the Equatorial Zone of the Ocean), Kiev,
"Naukova Dumka," 1976, 222 pages.
6. Sustavov, Yu. V. and Cfiernysheva, Ye. S., "Numerical Modeling of Internal Tides
in the Faeroe-Shetland Region," POVERKHNOSTNYYE I VNUTREPINIYE VOLNY, Sevasto-
pol', Izd-vo AN Ukrainskoy SSR, pp 168-177, 1978.
7. Sarkisyan, A. S., CHISLEI~i1YY ANALIZ I PROGNOZ MORSKIKIi TECHENIY (Numerical An-
alysis and Predi~tion of Sea Currents), Leningrad, Gidrometeoizdat, 1977, 182
pages.
- 8. Kochergin, V. P., TEORIYA I M~TODY RASCHETA OKEANICIiESKIKIi TECHENIY (Theory
and P4etliods for Computing Ocean Currents), i4oscow, "Naulca," 1973, 127 pages. ~
9. Gossard, E. E., Khuk, U. Kh., VOLPdY V ATMOSFERE (Atmospheric [daves), i~oscow,
"Mir," 1978, 532 pages.
10. Piagulin, V. V., Medvedev, V. I., Mustel', Ye. R. and Parygin, V. N., OSNOVY
_ TEORII KOLEBANIY (Principles of the Theory of Oscillat~ons), Moscow, "Nauka,"
1978, 392 pages.
11. Golubev, Yu. N. and Ivanov, V. F., "Influence of Bottom Relief and the ~-Ef-
fect on Long BarotroPic Waves," POVERKHNOSTNYYE I VNUTRENNIYE VOLNY, Sevasto-
pol', Izd. MGI AN Ukrainskoy SSR, pp 102-109, 1978.
12. Krauss, W., "On Gurrents, Internal and Inertial Waves in a Stratified Ocean
Due to Variable Winds:'Part 1, DEUT. IiYDROGR. Z., 29, 3, pp 87-96, 1976.
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GENERATION OF LONG WAVES IN OCEAN OVER LOCAL BOTTOri RISE BY ATMOSPHERIC WAVES
Sevastopol' TEORETICHESKIYE I EKSPERIMENTAL~NYYE ISSLEDOVANIYA POVERKHNOSTTIYKH I
VNUTRENNIKH VOLN in Russian 1980 (manuscript received 19 Jun 30) pp 108-117
[Article by V. F. Ivanov]
~ [Text] Abstract: With allowance for surface and bot-
tom friction layers a study was made of the
~eneration~of long waves in the ocean propa-
gating over a bottom rise through the action
of atmospheric waves. It was established that
in the surface layer the drift wave velocity
component makes a substantial contribution to
total wave velocity, especially for shorter
wave lengths. It is noted that the influence
of viscosity, the ~-effect and bottom relief
on the elements of the generated waves is re-
flected more significantly in the region close
to resonance and in the neighborhood of the
neaic of the underwater rise.
- Recently interest has increased in study ~f the variability of oceanological fields
caused by periodic oscillations in the atmosphere having characteristic periods
from several days to months [1]. Nevertheless, it is of interest to investigate
wave movements in the ocean in the range from several hours to a day, emphasizing
the upper surface layer in which intensive wave movements can exist. In many stud-
ies [2, 3J in which a study has been made of the influence of viscosity on the gen-
eration of surface and internal waves by atmospheric disturbances in most cases the
investigation was confined to the bottom boundary layer and the depth of the ocean
was assumed to be constant. An investigation of wave movements at the bottom with
allowance for viscosity in a basin of variable depth was made in [5].
In this article, by numerical methods, with allowance for the surface and bottom
friction layers, ~ study is made of the generation of long waves by periodic atmo-
spheric pressures in a homogeneous ocean of variable depth.
Assume that a plane ~aave of atmospheric pressure
~(kr -~t ) (1)
, pQ =po e '
ioo
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moves over the ocean. Here r= x cos y sin e; eis the angle between the direc-
tion of wave propagation and the x-axis. In a region of the ocean of a constant
depth it generates a wave in the form ~
e i(kr-6t~ ~2~
o '
which propagates over a local bottom rise having a domelike configuration [5]. The
bottom relief cf the underwater rise chs~iges in conformity to the formula
2,~X ?JIY ~X~~~ I.YI =L ,
- r1 !70 -H CQS ?e CDS 1,P1 when . Z ~3~
7
y ~y when ~X~ ~ , .
0
where ~1 and ~Q2 are the horizontal dimensions of the region at the base of the
_ underwater rise; H~ is maximum ocean depth; Hl is the elevation of the bottom rise.
We will investigate deformation of waves in the ocean generated by atmospheric
~ waves over the underwater rise and we w~ll study the structure of wave velocities.
The equation for the complex amplitude of a wave with allowance for viscosity, the
~-effect, bottom relief, wave shearing and no~mal stresses at the surface, ex-
pressed through atmospheric pressure gradients, is written i.n the form
? ~~ti
- 6s + 9 ~ l~l'a~~' ~ a~~ - ~ +
~ ? 6+ ~ [6*~ f
~ ~ ~
_ ~s, [ C ~ ] 1 ~ ~4,
+L C`l )'~*`y~ a _ ~ ~ y ~pa ,
~ , ~ ~p ~ ~d?~ eJ
where L= 2 CJsin ~ is latitu~e; C, 1=~+ Pa~g Pc ~
_ 6t~ , a ~ a
~ R i L ~1~. + ~ ~
~ '
~ ~v
_ and Ln is a complexly conjugate operator; ~1' and are the coefficients of turbu-
lent viscosity of air and sea water. In accordance with [4] we will assume that
they are equal to y' = 104 cm2/sec, y= 102 cm2/sec. In [6], within the framework
of the linear theory of long waves, for d> ~ the equation (4) was derived in which
we will neglect the secondary effects of the variability of e and H, associated
with viscosity, since their role is small for the deep ocean when the depth H> h,
_ where h is the thickness of the boundary layer (h = S~cY 1-1).
~ Knowing ~ and pa it is possible to compute the velocity components of the wave cur-
rent using the formulas
~ (5)
U~~ C~, Y~~~l'~~RCir ~
/1;1
. where
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_ c=-,,~ L p~ e~~-e>~� z_ ~R ~P~+~R~P~:~v e~>-r~l~ (y-:) ~
~ ~~~Q
(6)
L L ~
~ 2.Po ~6+~ ~
In formulas (5), (6) the first terms correspond to the drift velocity of tlie wave
current (we will denote the components of drift velocity by udr and vdr), the sec-
ond terms correspond to the gradient velocity and the third to bottom velocity.
The solution of system (4) is taken in the form

, ~ " ~ ~
, N
- where ~ is a disturbance introdu~ed by bottom relief;
i~('r ~ 8 ~
~ s ~a e
the solution with H= Ii~ = const. The complex amplitude of the wave S p with al-
lowance for
pQ=p,e~ (9)
- is found from (4), assuming the ocean depth to be constant,
~0,~0 (A-B) ~io)
po ~~*98~ ?
where
2 ? y~ -
,
~ ~ j ,
B~ k~~+`~ _N + y~ casB-~ (-1)
si~Bl`
a A 0 J1'
z~6+~ ~d? ~6~c n
Since beyond the local bottom rise the ocean depth is considered constant, the dis-
turbances introduced by bottom relief attenuate far from the underwater rise. Ac-
cordingly, the horizontal dimensions Ll and LZ (Ll> el, L2 } e2) of the investigated
_ region must be such that at its outer boundaries it can be assumed [5] that
_ ~ �0. (11)
At tlie internal points of the considered region we determine amplitude �rom equa-
tion (4). Substituting (7) into it, with (8) and (9) taken into account, we obtain
an inhomogeneous equation relative to C
d".t. ? H~~ - ~
) a~ - ~y ` L? ) +
~ ~ 6+C-~Y`~] ?dR C6~C-~) (12>
+L ' L'(H) m _ 9(B+C)+6 ~ ~B +C-A)po e~~~~
� ) n [ .Po
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where ? k.. - CCif~-i ~~S~R B[ a~ ~ r ay
. ~ Z[6?C-~)"~ l ~ ~ aX ~ ~ ay �
. n.~ J .
a ~ ~ ~
, ,
~ ,
8 ~ ~
i
~ ~ J
1 ~ ~ 2
y ~
r ~
I> Q d )
6 ~
i
i
/
1 ~ ~ 2 I~
~ -
y 2~ ~ ~ - -
v -
d b ~ ~
O,s > I ,
_ 2 1' n~~ ~ 2 2~ ,
~ `
- h ~ -
4 6 a ~ ~~01
B .
Fig. 1.
The derivatives in (12) are replaced by central finite-difference ratios and the
derived system of algebraic equations is solved by the Seidel iteration method.
Specific computations of wave elements were made for parameters of the problem
~ which then were not changed (~he linear dimensions were given in kilometers): ~1
, =,Q2 = 48; L1 = LZ = 30; H~ = 1; H1 = 0.94; 30�, the amplitude of the atmo-
spheric pressure was assumed equal to p~ = 100 Pa, the interval of the computation
grid was h~ = 4 km.
Emphasis was c~i an investigation of the influence of bottom relief on wave ampli-
_ tude and wave velocities and study of their spatial structure.
1Q3
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Figure la,b,c shows the dependence of the maximum amplitudes of the wave A and sur-
face wave velocities U and V on ~ for T= 2 and 0, computed for variable (curves
1 and 1') and constant (curves 2 and 2') bottom relief (here and in the text which
follows U and V are indicated in centimeters per second, A in centimeters, /1 in
'~,ilometers, Y and in square centimeters per second, T is in hours. Curves 1~nd
2 caere obtained taking into account all the terms in (4) and (5) and 1' and 2' for
' = 0. In Fig. l,a the curves 1' and 2' have been omitted because they are close
to 1 and 2. This small difference is aTtribu~tahas little effecthon waveWamplitude
tangential friction at the surface (t~
(by only several percent), but exerts a more signif icant influence on wave move-
- ments in the surface layer (Fig. l,b,c). In this case the greatest differences in
the amplitudes of Un, Vn from Ugr, V~r, computed for the cases V' = 104 and = 0
respectively, are observed an the slopes and at the base of the resonance curves.
It should be noted that for ~>a is the resonance wave length) the amplitudes
of velocities Un ~ Ugr, Vn ~ Vgr, whereas for 625 lj~ 0.953, where ~_~~l, 714. 76) . It can be seen (Table
2) that for a= 675 and a ~750 the influence of ~he ~-effect and bottom relief on
the amplitudes of waves and velocities is sma1Z and does not exceed 2-3% (the upper
lines correspond to the values of the maximum amplitudes A, U, V, coinputed without
taking the/g-effect into account and the lower lines with the ~-effect taken
into account). [Jith approach to resonance their role increases. For example, for
a= 705 and a= 725 the maximum amplitudes A, U and V change by 8-10% in compar-
ison with a case when the ~-effect i.s absent.
-02 o g2 v
,
~ �
~
. /
_ ~1i
1
/0,,~
, '
. ~
, ;
~ ,
~ .
~
,
~
, .
ti'
~50 .
~
~ ~ .
_ ~
~
_ . ,
GO
Fig. 3b.
- If 0, the amplitudes of the waves virtually do not change for waves propagat-
ing in forward ( e = 0�) and backward (e = 180�) directions, but the amplitudes
of velocities change by only 2-3% for ~1= 705 and ~l= 725. For these same wave
lengths, but with allowance for the `3 -effect, this difference is more signif-
icant and for both the wave amplitude and for velocities is approximately 15-17%.
In addition, if 0 and ?p, the amplitudes A, U and V~ are less for B= 0�
and greater for B= 180� than 3n the presence of a R-effect. In the case ~l>~1p
the opposite picture is observed.
With resonance i~) the role of the ~-effect is manifested in a decrease in
the maximum amplitudes of the wave elements. For example, the amplitude of the
wave is 32% less and the amplitudes of velocities are approximately 28% less than
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in the absence of the ~-effect. It must be noted that in a resonance cas.e even
with f3 ~ 0 the maxir~um amplitudes: of velocities. and uzaves virtually do not change
for B= 0 and B= 180�. The joint influence of the ~-2ffect and bottom relief in-
creases not only with approach to resonance, but also with approacfi to the peak
of the underwater rise.,Nevertheles~s, qualitatively the fields of amplitudes A~,
U; and Vg do not experience significant changes. In tfiis case the V field is more
intense to the right of the direction of movement of the wave both during its prop-
agation in a straight line and in the opposite direction. In a study of the role of
the ~-effect we assumed v' = 0 in order to evaluate better its influence on the
amplitudes of gradient wave velocities.
Figure 3 shows profiles of the velocities u and v, computed for the period T= 2
hours at the time t= 0(solid curves) and after a quarter-period t= 0.25T (dashed
curves) for a point situated at the center of the bottom rise (x = 0, y= 0). Curves
1 and 2 correspond to ~1= 200, curves 1' and 2' correspond to 400. The analysis
indicates that as time passes the profiles of the velocities u and v experience con-
siderable changes both in value and direction. For example, the velocity u for ~1 =
200 has a clearly expressPd maximum at a depth of 6.5 m at the time t= 0.25T, and
for 400 with t= 0. The v profiles behave similarly, but in contrast to u
the v velocities have a rather distinct maximum for ~1= 200 and t= 0(curve 1), and
also for a= 4Q0 and t= 0.25T (curve 2').
Summary
Allowance for surface shearing stresses has little effect on wave amplitude for
short periods (of about several hours) in comparison with the normal stresse~, but
is more important in computing the velocity of the wave current,especially in the
surface layer for shorter wave lengths.
The influence of bottom relief, and also the joint influence of the ~-effect and
bottom relief is manifested to the greatest degree near resonance and near the peak
of the underwater rise.
BIBLIOG1tAPHY
1. Lappo, S. S., SREDNEMASSHTABiVYYE DINAMICH~SKIYE PROTSESSY OKEANA, VO'LBUZH-
DAYEMYYE ATt40SFER0Y (Intermediate Dynamic Processes in the Ocean Excited by the
Atmosphere), Moscow, "Nauka," 1979, 181 pages.
' 2. Cherkesov, L. V., GIDRODINAMIKA POVERKHNOSTNYI~H I VNUTRENNIKH VOLN (Iiydrodynam-
ics of Surface and Internal idaves), I:iev, "Ilauka Dumka," 1976, 364 pages.
3. Levkov, N. P., "Dissipation of Surface ~daves Generated by Periodic Atmospheric
Disturbances, P40RSKIYE GIDROFIZICHESY.IYE ISSLEDOVANIYA (t4arine Iiydrophysical
Investigations), No 4, Sevastopol', pp 76-83, 1973.
4. Sarkisyan, A. S., CHISLENNYY ANALIZ I PROGNOZ MOR.SKIKH TECH~NIY (Numerical An-
alysis and Prediction of Sea Currents), Leningrad, Gidrometeoizdat, 1977, 182
pages.
l0B
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8. Golubev, Yu. N. and Ivanov, V. F., "Deformation of a Long IJave Over an Under-
water Mountain," MORSKIYE GIDROFIZICIiESKIYE ISSLEDOVAt1IYA, No 1, Sevastopol',
pp 32-43, 1978.
6. Ivanov, V. F., "Influence of Viscosity and the r-Effect on the Generation of
Long Waves in the Ocean by Atmospheric Waves," in this ~:ollection of article~,
' pp 99-107.
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INFLUENCE OF FLUID VISCOSITY ON WAVE RESISTANCE TO SYSTEM OF NORMAL STRESSES
- DISTRIBUTED IN SEGMENT
Sevastopol' TEORETICHESKIYE I EKSPERIMENTAL'NYYE ISSLEDOVANIYA POVERI~HNOSTNYKH
I VNUTRENNIKH VOLN in Russian 1980 (manuscript received 19 Jun 80) pp 118-123
[Article by L. G. Yeremenko]
[Text] Abstract: In tliis article zne wave resistance
of a viscous fluid is represented by the L. N.
Sretenskiy integral. An asymptotic solution is
obtained in the case of an arbitrar~ viscosity.
I:no~m solutions are obtained in the case of van-
ishing viscosity. Pdumerical computations indi-
cate the presence of such regions of paraneters
in which even a small viscosity gives results
differing substantially from the results for
an ideal fluid in these same regions of the val-
ues of the parameters.
In a linear formulation we solve the problem of computing the wave resistance R
of a viscous fluid covered by a viscous film with the steady m~vement of normal
pressures p~, uniformly distributed in a segment [1, 2], over the surface of the
f luid
~ ~ X - ~ ~ /d+~d V ~ [l~l V VaO+ ~1)
~ R: ~ p (z) �'x dx; (2)
~p
Wltlt Z = 0
-P ~2 pc ~ +.P9 �c �'~a iR, ( X~ ~ ( 3)
'�sP?P9z. f~ ? = 0,
(5)
- u~,~Z; .
iio
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vX, vZ, ~O-~0 when z-~ - oo ; (6)
dY ~
-
dXp' vx ~ rZ~P~ ~ when ~ x~-~oo, (7)
Q , Ixlse ,
p* (X)=
_ o , i~l~~~ . c8~
Here P is fluid density; ~is the kinematic viscosity coefficient; u is the velo-
city of movement of normal pressures; /~f is the density of the surface film; vX
and vZ are the velocity components of the fluid; OiC is reduced surface tension
� [5]; z is the rise of the free surface;,0 is fluid pressure; P is the dynamic com-
ponent of fluid pressure. ~
The origin of coordinates is set on the undisturbed fluid surface. The oz-axis is
directed ~~ertically upward.
The system (1)-(8) is solved by use of an integral Fourier transform of x. Invert-
ing, we obtain the following integral expression for rise of the free surface:
~ > S - e ~~XI~I P C~)d~
~ i ~ .~~2 ~ ~ 2 - ?+C91 ~'I~,~ I~f')-y~ 3~Z I~I~ v~'Z - u ' c 9 ~
,o~~"~=~ ~P(X)e cfxdz.
' After some transformations the integral expression of wave resistance is represent-
ed in the form .
1Q2 f i siar B I~I d
9
~ ~p ?~~'t i~u z+(91~1+,~1~'1~1-4 ~rv-i~u
_ clo)
- ~ s d/~ �
Investigating the expression for ~q ~at the extremum,
y r ~ -c~ �
t4?s lc~~~=_~~~~ t+9~~ ~ ~ ~ ,
it can be demonstrated that for all E(- oo < E< +no) the following evaluation is
correct
+ 2
~Q~ ~~s~y y: .
a
= I3y virtue of this evaluation the wave resistance (10) can be written in the form
of a converging series [3, 4)
~ ~ �O i aia: ~ ! ~~I � ~ d~ (11>
R = ~ R *R f ~1 V t -c,~cl~ +(9 ~~+~6 ~ ~ + R"' .
_ ~.c a p.o.
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We will make a replacement of variables gv/u2 and then urrite the main term of
the expansion . . ~ ~
R=_ 242~ isi~tz v/Frzdv +
,~1~ o v 4 9~ v- 4 i~- ) v_ v+
p p? i sl : v F~: _d v r (12)
?~j" f r-~+ 4i v t-v +f
P9 0 ~
where ~ 1
; fr=~; l'~~=-uA-~- %
WE is the Weber number, Fr is the Froude number.
We will break down the integrands (12) into simple factors
R__ ~~r' � dv? > ~ cas Zv/Frt dY+
~p9j ~ ~-~~t ~ o ~ ~
~
+ f �'dv ~ 1 ~_eo~P?r ~r:dv ~ ~
_ -
- ~'o~"r" ,ri ~ ~ ~~=~~*d~~�i-~'~~f ~ ~
J / ~ .
,
? 1 ~osl.r/f 'r~,dr+~, r~
~v d~_ 1 cat 2ti?ifr! dv . (13>
� . Y K� J y-~� y~ y- v�
. / p J 0 f~ J
Here v~ and v~ are the roots of the equations respectively:
49~t rj-(4~ ~-!'~rj-v~J-O, 4~i vs+(4l~+1'~ v?-"v+!_0.
Using expressions 3.722.7, 8.232.2 from [6J, we finally obtain
,0 ~ f j..~_~~
L ~
R=,~p91 fa' f(it~ . r,� v,�t-/) ~ rY
+eps 1 t v~ ~1~- v~~+iia~f rt ~~/c~~~ ? (14>
a v ~fr ~
. > j 2 ~ +
~ ~ >t 9~ _ + 8 v � -f ~ l C ~e~ f
r t - l a ~
r.> { ~ ~ )
~~z
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-~~C-~o ~~-~)+Jl~C~ ~~)[s[C~ Y,
. (14)
The residual of the series can be represented in ttte following way:
Z'Qt i s~~ t vl ~ d v.
~N AP9-~. 4`~ t -41 l~vt ~/'r~-~v~ t> v !-4
We will evaluate RN with respect to absolute value, assuming N= 2k. Then IRNI
+ 2I~ I, where _ _ . _ _
I�� ~41~ ~ 4~z v~+ >+~d r
~~~~J~y 4g'?v3+!'vz-v~>~1+16~ v4 ~ 1v3t>+~'v -4g'z~�).
It can be shown [5] that
~ 4 9e Ys + Y+ J I+/'V:
<
(y s~? v~~ /'v 1- v+ +l6 ~ ~ 2v3~>.~ /'r 1- ~ v y ? ~
Then for I~1 ~ we obtain ~ y,~
. `~J Ey v l+/'vz) d~'
0
We will break down the integration interval into two parts [0, A] and [A,~ )
~ ~ + ~r~ ~f dr
IJ1I 2~ YEy ~1~!'Y ~ l~l L J Y~' /'~'/~Yt~
/ A
and in the segment [A, oo ) we evaluate Iq~ d v ~ 1 ,p.~,r/t-4 ~
J ~ f ~ Y~~~-S'~IQ~Y'~ ~ ~~.?.f 2
~ .
~ 2~2` O 27~` /
We will select A from the consideration that both terms on the right-hand side of
the latter inequality are of the same order of magnitude
~ t-t ~3C~'~yJ~=r~t , A~~~~1~1s/~ ~ ~ A=/" ~ R.y ~ t,t- ~t- '
Thus, we obtain the following evaluation of the residue of the series:
. _ _ _ . _ _ _
. IRRI y~ 3~ ' ~ ' '
Numerical analysis of problem. In the case of steady movement of the normal stress-
' es distributed in a segment we made computations of the total resistance using for-
muia (14) for the cases ~6 = U and 0, and also using the L. N. Sretenskiy for-
mulas [7] for the wave resistance of an ideal fluid and for the wave resistance
of an ideal fluid with a correction for viscosity
~ yQt ~ ~ yPt ? 9� 1
_ R~ ~9 sin1 f-~ , R-R~ P9 ` G'
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The computations were made for different Fr and WE numbers. It follows from the
computations that the influence of viscosity in essence is expressed only for
velocities of movement greater than 20 m/sec. The presence of a region of a
sharp rise and fall of the wave resistance of a viscous fluid is of interest,
whereas for an ideal fluid a tendency of wave resistance to a limiting constant
value is characteristic. The influence of a surface film is important only in
the rise region.
R ~ . -
~725 ~ .
~1 5T
_ ~
_ ~ ~s ~ /
. ~
~z5 /
_ R~7 /
~J7 ~.47
0,75 ~
~ . 1
g15 ~ ~
~ -
3 9 !d Zl 17 ,~1 J9 4J S/ cc
Fig. 1. Wave resi~tance: of moving pressure
o ~ Ixl s~ ,
P,~=~ 0, Ixl~e '
for a viscous fluid; for a viscous fluid with allowance for surface tension;
for an ideal fluid.
For velocity values less than 20 m/sec the computed values of wave resistance for
a11 four expressions coincide with a great accuracy.
L. N. Sretenskiy [1] derived his formulas by the method of expanding the integrand
of wave resistance into a series in powers of viscosity near vanishing viscosity.
The correction for viscosity obtained by ttxis method was insignificant. In this
study we have investigated this same L. N. Sretenskiy integral, but have obtained
114
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- a different asymptotic expansion of this integral for arbitrary viscosity, from
which the L. N. Sretenskiy solution is ohtained as a special case.
The curves presented in this study sfiow that it is possible to indicate the zones
_ of values of the parameters for which the influence of viscosity is important.
A clarification of the influence of different factors on wave resistance is neces-
sary in developing means for lessening it.
BIBLIOGRAPHY
1. Sretenskiy, L. N., TEORIYA VOLNOVYKH DVIZHENIY ZHIDKOSTI (Theory of Wave Move-
- ments of a Fluid), Moscow-Leningrad, OPZTI NKTP SSSR, 1936, 108 pages.
2. Kochin, N. Ye., Kibel', I. A. and Roze, N. V., TEORETICHESKAYA GIDROZiEKHANIKA
(Theoretical Hydromechanics), Vol 1, Moscow, Fizma.tgiz, 1963, 445 pages.
3. Potetyunko, E. N., "Asymptotic Analysis of Surface Waves af a Viscous Fluid
With Small and Large Times," DAN SSSR (Reports of the USSR Academy of Sciences),
Vol 210, No 5, pp 1040-1042, 1973.
4. Potetyunko, E. N., "Asymptotic Analysis of Surface [Javes on a Viscous Fluid
of Infinite Depth," IZV. SKNTs VSh. SER. YESTESTV. NAUK (News of the Northern
Caucasus Scientific Center of Higher Schools. Natural Sciences Series) [Expan-
- sion Unconfirmed], pp 71-74, 1973.
5. Yeremenko, L. G. and Potetyunko, E. N., "Analysis of Waves and Wave Resistance
of a Viscous Fluid With Allowance for a Surface Film," GIDROMEKHANIKA VYSOKIKH
SKOROSTEY (Hydromechani~~s of High Velocities), Leningrad, "Sudostroyeniye,"
313, pp 145-153, 198~.
6. Gradshteyn, N. S. and Ryzhik, I. M., TABLITSY INTEGRALOV, SUt~IIri, RYADOV I PROIZ-
VEDENIY (Tables of Integrals, Sums, Series and Products), Moscow, Fizmatgiz,
1963, 1100 pages.
7. SretEnskiy, L. N., "Waves on the Surface of a Viscous Fluid," TRUDY TsAGI
(Transaction~ of the CF~ntral Aerohydrodynamics Institute), No 541, Moscow,
pp 1-34, 1951.
8. PROBLEMY PRIKLADNOY GIDROMEKHANIKI SUDNA (Problems in the Applied Hydromechan-
ics of a Ship), Leningrad, "Sudostroyeniye," 1975, 251 pages.
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~
LONG SURFACE AND INTERNAL WAVES GENERATED BY NONAXISYrII~tETRIC INITIAL DISTURBANCES
Sevastopol' TEORETICHESKIYE I EKSPERI2IENTAL'NYYE ISSLEDOVANIYA POVERKHNOSTNYKH I
VNUTREI~IdIKH VOLN in Russian 1980 (manuscript received 18 Mar 80) pp 124-135
[Article by S. F. Dotsenko and B. Yu. Sergeyevskiy]
[Text] Abstract: In a linear formulation the authors
examine the process of development of spatial
long waves in a two-layer fluid caused b;~ ini-
tial displace:nents of tlie free surface and inter-
faces: of tfie layers. A study was made of the influ-
ence of asymmetry of the initial disturbance on the
developing unsteady surface and internal waves. The
rises are modeled by a set of a finite number of
identical (but displaced relative to one another)
axisymmetric displacements of a special type, for
which the solutton of the problem is expressed
through elementary functions. It is shown that for
extended initial disturbances waves radiated perpen-
dicularly to the greatest axis of the rise are dom-
inant. A comparison witli botli plane and a:~isyia-
metric waves is presented. Regions for which the
spatial waves are close to plane waves are deter-
mined.
The process of evolution of nonaxisymmetric initial displacements of the free sur-
face and the interfaces of layers of a two-layer fluid is examined. The influence
of the asymmetry of the initial disturbance on the developing surface waves was in-
vestigated earlier in jl, 5, 6].
1. A basin of the constant depth H, filled with an ideal incompresiiible two-layer
fluid, unbounded in horizantal directions, is examined. The thickness and density
of the upper layer are hl and p 1 respectively; for the lower layer, h2 and~0 2,
the corresponding parameters are ~~
~a~ CR~ i~ } i / \ c c
Then the integrals ~lk in (1.3) are computed using formulas 6.554(4), 6.623(2)
from [3J. Finally we find
_ _ Z._ - 1- _
~r~ ~Aie~ais~i' ~u~'~~'`12e~~i~s~i"i1' .
r~ ~ (1.6)
~ Z ~ 1 ~ .
- ~,t'A2ECZ1ei ~.Bsi;~;Z~ ~iz'~'~i~ L ~d ~d~~iB'~;Z)szj ~
where ~
sy y~ j Ze~2 ~os(Z c~~11 + N~ s~a~2 ~ r~ ~f
4.
l ~ '
T~ ~ M~~ * N?; M~ Z+ e2= t Z; ~v.~ . ze. u~ t;
the angles ~i~ E[0, 1i,'J also satisfy the system of equations
_ i/2
~ si~ c~~~ ~,N.~ T~ , cos y~~ = M~j Tj �
An analysis of the process of formation and propagation of long waves for this
case is contained in [4].
The solution of the axisy~netric problem makes it possible to study some features
of development of spatial waves in more complex cases when the initial displace-
ments of ttie free surface and interface of fluid layers do not have cylindrical
symmetry. We will use, as in [1], where the case of a homogeneous f luid was ex-
amined, the very simple method for modeling of such wave processes. It rests on
the algebraic addition of a finite number of identical initial axisymmetric dis-
~lacements of the form (1.5), whose centers are situated at different points ~xi~,
yi~) in the horizontal plane (i = 1 is the undisturbed free surface of the fluid
z= 0; i= 2 is the interface of the layers z=-hl)�
R~
Assum~ that ~ ~ (
. ~i~~ A~ ~oi ~ o i ~ ~ ' (1. 7 )
- i
~i
R/.~ (X-Xr~~Zt (Y~Y~~)Z , /lo~=ntaz{~ ~ioi\Rf)}~
~ ~
X~Y ~,~t
where ni are natural numbers; i= 1, 2. Then, due to the linearity of the problem
from for.mulas (1.3), (1.5)-(1.7), we obtain
Z (1.8)
t \ Rs Aoi ~ ~ ~.k ~Rt~ . t~ .
C' y' ~ .
~.-R~ k~/
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With t= 0 the maximum values of displacements of the surface and interface of the
fluid layers are equal to R1 and R2 respectively.
- The simplest case is that examined below when the points (xi~, yi~) lie on a
straight line at an identical distance Q Li from one another. jJithout limitations
on universaltty it can be assumed that this line coincides with the Ox-axis; the
point (xi0, Yi~) is at tne origin of coordinates. Then in formulas (1.7), (1.8)
x�� = j 0 L, y = 0(i = 1,2; j= 0, �ni) the displacements ~ i(x,y,t) are even
i~ i i,j
functions of x~:nd y. This makes it possible to limit ourselves to ~~n analysis of
waves only in the region x> 0, y~ 0.
The asymmetry of the initial rise of the indicated type is conveniently evaluated
by the parameter ? i, equal to the ratio of the characteristic length 2Lli of the
rise to its cttaracteristic width 2L2i( ~ i= L1iL2i1~' If ao the conditional boun-
dary of the initial rise we take the isoline on w~iich ~ i(x,y) = O.lAi, then as
Lli it is desirable to ~ake the maximum distance from it to the origin of coordin-
ates, and as L2i it is desirable to take the minimum distance from it to the ori-
gin of coordinates.
- A further analysis of waves was made using the formulas (1.3), (1.6). The valt~c~s
hl, h2, ~ were taken equal to 0.8, 4.2 km, 10-3 respectively, as is characteris-
tic for the ocean floor. It was also assumed that ~ 1= Q 2= 52.4 km, L1 Ll= d L2
= 30 km. In all the figures presented below the distance is measured in kilometers.
2. Assume that A1 = l, AZ = 0. In this case at the time t= 0 the free surface of
the fluid deviates from the position of equilibrium; the interface of the layers
is horizontal.
In an axisymmetric case from (1.3), (1.4) we obtair. approximate expressions far ~l
and ~2
Z 1 /
j~~ _ ~/-Eft2 y"z.\ * ~ ~11hr- ~Z , ~2= fL2// 7~ - ~2 y ~2 ' (2.1)
J
where a �
r~~- ~~rcosr ct f, (r~j,(rR~ dr ; r~2= ~ rcosrccZtf(r}7~rR)a'r, ~2.2~
0 0
and nl coincides with the precise expression for the rise of the free surface of
a homogeneous fluid.
It follows from formulas (2.1), (2.2) and the results of [4] that the initial axi-
symmetric displacement of the fluid surface leads to the formation of surface and
internal waves attenuating with time. The first terms in formulas (2.1) corres-
pond to a surface wave; the second terms correspond to an internal wave. A surface
- wave is slightly distorted due to inhomogeneity of the fluid and in the process
of its propagation causes a displacement of the interface of layers of the same
sign which is less by a factor of Hh21. The velocity of propagation ~1 is evi-
dently close to c. The amplitude of th~ developing purely internal wave is maximum
at the interface of the layers; at the free surface it is H(~ h2)-1 times less
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and the velocity of its propagation is u2 � c. Due to the lineaY�ity of the ini-
tial mathematical problem the mentioned general properties of unsteady waves in
a two-layer fluid are correct for nonaxisymmetric initial displacements of the
type (1.7).
. -
- _
~ t=5 ~eax min \
' S ~
\ S
19 >S
SS
_ ~ ----~p~
~ ~ t +1S Mas
_ ~
S 1~
~ ~
100 45 40 ,30 yp ~ S .iS
� \ ~ ~ _ _-1 ~
~
~ ~ ~
- s ~0 ~ ~D
- 0 -5 ~ ~0 - ~ '
t'~ ~g min
~ f0 S ~ 10 S\`~~
4 40 ~ ~ ~ ~
\
Z00
5 ~
-S ~ \ -JO ~ ~
_ , ~q-lS ~ ~
- ti a ? a
0 ZQ7 4A? 0 ~ 40D
Fig. l. Form of free surface and lnterface of layers in initial stage of wave de-
velopment (A1 = 1, A2 = 0): a) isolines 102 zl; b) isolines 102 z2.
- The process of development of unsteady surface and internal waves in the case of
- an initial displacement of the free surface of the fluid with nl = 7(L11 = 295
km, L21 = 145 km, y1= 2.04 is illustrated in Figures 1 and 2.
Thus, the initial displacement of the free surface of the fluid in the process of
its evolution leads to the formation of a spatial surface wave T~1 attenuating with
time (Fig. l,a) and disturbances of the interface of the layers ~2 (Fig. l,b),
representing (see (2.1)) a superpositioning of the developing internal wave and
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the "respunse" of the surface wave at the density jump. In the initial stage of
wave development there is a subsidence of the central part of the rise ~1. The
maximum ~1 values with t~ 0 are situated on the Oy--axis and move along it in oppo-
site directions with velocities close to c.
.
. _ _
_ hours
Z00 \ \ \ ~
- 7J ~
-m'S - - 0
0
t ~15 Y
, \
-s - -5
ZDO \ ~ ~ 0.
~
?S ~
\ '
i~~-_~~ \
1 ~
~ ~ 1~ -S -S ~
0
- \ t=2�Y hours
~ ~
S ~
S
~
~
' ?9 ~
1Gb
~m
~ ~ ~ _S \ ~
S , _S
1,~ ~ ~ -1k 1
- 0 a70 4m D ?A9 4A7
a ~
Fig. 2. Form of free surface and interface of layers with large t(A1 = 1, A2 =
0): a) isolines 102 ~2; b) isolines 105 ~1�
At a definite moment in time t= tp (5 < t~ < 15 min) at two points txp on the Ox-
axis (x~ > 0) the di~placement ~1 becomes equal to zero and thereafter two nonin-
tersecting regions with ~1 ~ 0 are formed, these being situated symmetrically
relative to the Oy-axis. With an increase in t there is a broadening of these re-
gions and their joining into one (t = 15 min), taking in the origin of coordin-
ates. Points with a maximum depression of the free surface with t< 20 min are
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displaced along the Ox-axis in the direction of the origin'of coordinates and then
move along the 0y-axis in opposite directions, following the corresponding iso-
line of maximum amplitude. At an adequate distance from the origin of coordinates
the ~ 1 isolines constitute closed curves, as is characteristic for axisymmetric
waves.
In the pro~ess of submergence of the central part of the initial rise of the free
. surface ~ 1 the generation of an internal wave occurs. Initially it ~.s a region
of negative ~ 2 values; I~ (
max
x,y 2
- is situated at the origin of coordinates (Fig. l,b). In addition to a local depres-
sion of the density jump, an internal rise wave is formed which corresponds to the
' main surface wave and moves witli the velocity of long waves in a homogeneous flu-
id. It is the "response" of a surface wave, as was mentioned at the beginning of
= this section. Its amplitude, as mentioned earlier, is Hh21 times less than the am=
plitude of the surface wave.
In accordance with Fig. l,b, in the neighborhood of the origin of coordinates in
the initial stage of movement of the fluid a region of negative ~2 values is form-
ed, elongated along the Ox-axis. In the process of its evolution, when the main
disturbances of the free surface become small, purely internal waves are formed
. (Fig. 2,a); the velocity uf their propagation is u2 c. It follows from a com-
parison of Figures l,a and 2,a that the spatial structure of the surface and in-
ternal waves qualitatively has much in common if -~2 is considered, rather than
~2'
In the process of propagation of this internal wave it exerts a reverse effect on
the free surface of the fluid. The developing surface waves (Fig. 2,b) constitute
a"response" of the internal wave at tlie free surface of the fluid. Their ampli-
tude is less by a factor greater than 103 than the maximum vertical displacements
within the flu3.d. The spatial structure of ~1 and ?.2 with t> 15 hours is different;
especially the positive ~1 values are localized in an annular region, which was
not observed for internal waves.
At one and the same moment in time t the amplitude of a purely internal wave is
slightly dependent on the relative density drop but the velocity of its prop-
agation and the amplitude of the surface waves caused by it decrease�substantially
with a decrease in ~ .
Thus, the initial displacement of the free surface of a two-layer fluid in the
course of its development can generate internal waves whose amplitude is of the
same order of magnitude as the amplitude of the initial disturbance. Such waves
_ attenuate considerably more slowly than surface waves and are dominant after de-
= parture of the main surface waves from the region of the initial displacement of
the free surface of the fluid. In all cases for initial displacements o� the
fluid surface elongated along the Ox-axis the waves of greatest amplitude are
radiated in the direction of the Oy-axis. Such a directivity of radiation of un-
steady waves is observed even with ?1 valu~s extremely close to 1.
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Assume that A1 = 0, A2 = 1. In this case at the time t= 0 the free surface is
horizontal, the interface of the layers is displaced from the position of equil-
ibrium.
In an axisymmetric case from (1.3), (1.4) we o3tain approximate expressions for
~ 1 and j 2 _ -
jl=EhZH-~ (~,_~2~ ~ lz~~~2~-2~~~~I-t'ftZH~Z~f~Z , (3.1)
where ,e
~~a ~rcos rctf (r)Jo ~r~dr; f~t~~rco~r~=tf J,r dr.
~C~ ~s.2>
D ~
It follows from formulas (3.1), (3.2), (1.7) and the results in [4] that the ini-
tial axisymmetric density jump leads to the formation of surface and internal
waves. The first terms in formulas (3.1) correspond to a surface wave; the second
terms correspond to an internal wave. The amplitude of the surface wave is of the
order of 0( and therefore is substantially less than the amplitude of the in-
ternal wave. It is evident that j~ 2 N E2.
The process of development of unsteady surface and internal waves in the case of
initial displacement of the interface of layers in the form (1.5), (1.7) with n2
= 7(L12 = 295 km, L22 = 145 km, 1~2 = 2.04) is illustrated in Figures 3, 4.
The initial displacement of the interface of layers of the fluid (Fig. 3,a) in
the process cf its development generates surface waves of two types: first, pure-
ly surface waves propagating with the velocity c(Fig. 3), second, disturbances
of the free surface, caused by inhomogeneity of the fluid (Fig. 4,b), propagating
with the velocity of internal waves u2. The formation of surface wavES of the
first type caused a slow subsidence of the central part of the initial displace-
ment of the density jump. These waves constitute two regions of positive ~ 1 val-
ues propagating in opposite directions and having maximum amplitude values on the
Oy-axis (Fig. 3). Thereafter the deformations of the initial rise ~ 2 and the free
surface increase at the origin of coordinates, which lead to the formation of
slowly evolving internal waves (Fig. 4,a), which correspond to qualitatively sim-
- ilar surface deformations of the fluid of opposite sign, representing the above-
mentioned second class of surface waves of small amplitude (Fig. 4,b). All the
considered waves are essentially three-dimensional. For initial displacements of
the interface of layers elongated along the Ox-axis the waves of greatest ampli-
tude are radiated in the direction of the Oy-axis.
It is clear from a comparison of Figures l,a, 2,a, 3,a and 4,a that the spa.tial
structure of the fields of purely internal and purely surface waves has much in
common, although these classes of waves have e~sentially different propagation
velocities. This conclusion can also be drawn from formulas (2.1), (2.2), (3.1),
(3.2). In accordance with (1.7), the spatial structure of the mentioned surface
waves is described by the sum of the integrals ~1 or ~ 1, for internal waves
L2 or � 2. It is evident that with ~1 2, as a result of the replacement _
u2c-lt the integrals ~2 and � 2 assume a form similar to ~ 1 and ~ 1.
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t~0 t=5:~tzx min
200
\ \1,0 ~~5 6;~
~s ~
s ~ . ~ \
/0 ~
100 J9 ~
~ti
_ 0 _ ~ 5 ~ ~ T
a a ~7 ~
t~/5 c~i~~c
~
t =1 aa~a min ~ '
~717 " ~
~
.
. \
~ \
0 ~ ~
~ _ ~
0,3 ~ ~
~ 0,3 ~
100
- Z -~10
\
- ' 1
~ 200 400 0 ~ 2A~ 1+00
~ b 2 d
Fig. 3. Form of free surface in initial stage of wave development (A1 = 0, AZ = 1):
a) isolines 102 ~2; b, c, d) isolines 105 ~rl' ,
The amplitude of the internal wave to a considerable degree is dependent on the
initial parameters of the problem and this dependence is particularly important
in the case of large t values. A change in hl, h2 and leading to an increase
in u2, not only accelerates the development process, but also decreases the ampli-
tude of the internal wave at a particular moment in time, all other conditions be-
ing equal. ~
4. We will examine the influence of asy~netry of the initial disturbance on the
amplitude of surface and internal waves. We will assumz that Ai = max zl(x,y,t)
x,y
with A1 = 1, A2 = 0 and A2 = max~2(x,y,t) with A1 = 0, A2 = 1. The di = 1 value
x,y
corresponds to axisymm~netric waves and ~i =�O corresponds to plane waves.
The computed expressions Ai = Ai( y i,t) are represented in Fig. 5. The ~i values
are indicated on the corresponding curves. It therefore follows Chat Ai(l,t)< Ai
(~l 2, t) 1. Accordingly, for initial rises of the free sur-
face and the interface of the layers elongated along the Ox-axis, with their
- identical width, the amplitudes of the developing waves are greater than in an
axisymmetric case and increase with an increase in ?i. It is evident that
Ai t) N 0. 5 with large t, but Ai i, t)-+0 with t-+
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t=5+~ hours
1Gfp ' f, \S\~
\ -5 3 \
~S
~
6~0 ~
_ � ~ t-s~ . .
f ~ ~2 ~ 5 \
14 -A?
20D
- ,
. ~ ~
~ ~ i
~ Z � . t=24~ths rs ~
� \ -2\
>0 5
~ "
200 ~
~ - -'0-~`\ .
~ ti`' ~
i
D 1Q0 a ~ y~ 0 ?Q7 d ~
. Fig. 4. Form of free surface and interface of layers with large t(A1 = 0, A2 = 1):
a) isolines 102~2; b) isolines 10571�
In conclusion we wi11 examine the problem of regions in the plane x0y for which the
form of the spatial wave differs little from the form of a plane wave. We will as-
sume that for such regions
- ~~i' ~'y t~ '~~(y,t) ~ sO,lAi , (4.1)
where 2~ is the displacement corresponding to a plane wave (?i Th.e results
of computations of the boundaries of these regions for an internal wave (i = 2) in
the case A1 = 0, A2 = 1 are shown in Fig. 6(n2 = 7). The inequality (4.1) is sat-
isfied in one (in the initial stage) or two time-dependent regions of ell'.psoidal
shape situated symmetrically relative to the Ox-axis (Fig. 6). With an increase in
t they withdraw from the origin of coordinates with the velocity u2, decrease in
size mon.otonically along the Ox-axis, but clian~e little in tlie direction of
the Oy- axis. At some moment in time this region ceases to exist. With an increase
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'in v2 the size of the ~egions along the Ox-axis in the time interval during which
they exist increases.
y
~ S 10 15 t~~v \1 '
+ t minutes 6 ~
a 1 .
~ `
~ ' s ~ .
1
~ . 1Q7
~1--__ ~ \
~ --1,7'-- 1
- ~ '
~4 ' '."s~ � - � ~ ~ 1
' :
~=Z . . . ~ ~ ~Zz` ` ~ ~
4 ~
0 ~Q 1~J L` 4 hours
. 0 ' SO 1Q~ x
Fig. 5. Dependence of Ai on t for differ- Fig. 6. Boundaries of regions for in-
ent ?i values. ternal waves within which the inequal-
ity (4.1) is satisfied. The curves 1-6
correspond to the values t= 0; 1; 2;
5; 10; 15 hours.
Similar results are correct for surface waves ~1 caused by initial displacement of
the free surface of the fluid [1].
~ BIBLIOGRAPIiY
1. Dotsenko, S. F., Sergeyevslciy, B. Yu. and Cherkesov, L. V., "Long ~Javes From
Nonaxisymmetric Initial Displacements of the Free Surface of a Fluid," EVOL-
YUT'SIYA TSUNAMI OT OCHAGA DO VYKHODA NA BEREG (Evolution of Tsunamis From the
Focus to Emergence on Shore), in press.
2. Sekerzh-Zen'kovich, T. Ya., "Propagation of Initial Disturbances Along a Free
Surface and Along the Interface of a Fluid Consisting of ~ao Layers of Differ-
ent Density," TRUDY MGI (Transactions of the Marine Hydrophysical Institute),
Vol 17, Moscow, Izd-vo AN SSSR, pp 48-58, 1959.
3. Gradshteyn, I. Z. and Ryzhiky I. M., TABLITSY INTEGRALOV, SUI~! RYADOV I PROIZ-
VEDENIY (Tables of Integrals, Sums of Series and Products), Moscow, Fizmatgiz,
1963, 1100 pages.
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4. llotsenko, S. F, and Sergeyevskiy, B. Yu., "Axisymmetric Long Waves in a Two-
Layer Fluid From Initial Disturbances, POVERKHNOSTNYYE I VNUTRENNIYE VOLNY
(Surface and Internal Waves), Sevastopol', Izd. MGI AN Ukrainskoy SSSR, pp
119-128, 1979.
5. Sen, A. R., "Surface Waves on Fluid of Finite Depth Due to Arbitrary Surface
- Impulse and Elevation," J. TECHNOL., Vol 4, No 2, pp 105-118, 1959.
G. Chaudhuri Kripasindhu, "Waves in Shallow Water Due to Arbitrary Surface Dis-
turbances," APPL. SCIENT. RES., Vol 19, No 3-4, pp 274-284, 1968.
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INTERNAL WAVES FROM INITIAL DISTURBANCES IN A TWO-LAYER FLUID
Sevastopol' TEORETICHESKIYE I EKSPERIMENTAL'NYYE ISSLEDOVANIYA POVERKHNOSTNYKH I
~INUTRENNIKH VOLN in Russian 1980 (manuscript received 3 Apr 80) pp 136-142
~ [Article by A. A. Novik]
[Text] Abstract: A solution of the plane problem
of gravitational waves arising at the hori-
- zontal discontinuity of two flows of fluid of
different density is obtained and investigated.
We will examine two flows of an ideal fluid situated one beneath the other and ex-
tending without limit in a vertical direction. Assume that the lower flow has the
velocity cl and the density ~O ; the upper flow has the velocity and density c2
and 2 and Pl>~2. At the in~tial moment in time t= 0 the fluid particles are
imparted any additional velocity and the initially horizontal discontinuity changes
somewhat. It is necessary to find the form of the discontinuity at any moment in
time t> 0.
We will cite the equations and boundary conditions of the problem of internal
waves. We will use the horizontal line of separation of flows in an undisturbed
state as the x-axis, whereas the y-axis is directed vertically upward. We will as-
sume that the ~ coordinate system moves in the plane of motion in a horizontal
direction with the constant velocity
r p~`j
4 f ~ � (l~
. P~ * P2 " ~
Then the linearized boundary conditions at the discontinuity assume the form [2]
~ a~, ~ P~C~.-~Z~ ay, t 9~ dyi a~t +92',
.P' ~ ax ~ ~ at ax J c2>
at p, t p2 p~ p2
. a7 + P~ C~~' ~ i-, c3~
o~ p,,- pZ aX a Y
. d? cZ~ g? ~ dtp: c4)
, at P,~ F~ ax ay
iza
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Here g is the acceleration of Kravity; rf(x;t) is the deviation of the discontinuity
from the position of equilibrium; ~1(x,y;t) and ~ 2(x,y;t) are the potentials of the
velocity of wave ~ovement of the lower and upper fluids; the derivatives of the
functions ~ 1 and ~ 2 are taken with y= 0.
The problem of internal waves is usually formulated in the following way. It is nec-
essary to find such a solution SP1 of the Laplace equation in the lower half-plane,
becoming equal to zero together with the first-order partial derivatives with y=
and such a solution SG2 of this same equation in the upper half-plane, having
a similar behavior with y=~, which on the axis y= 0 would satisfy the two boun-
dary conditions obtained from (2)-(4) by exclusion of the function , and also
some initial conditions. With known velocity potentials the discontinuity equation
is found using formula (2) by simple differentiation.
2. An analysis of the equations and tlie boundary conditions makes it possible to
simplify the problem somewhat.
First, the right-hand sides of the kinematic conditions (3) and (4) are the normal
derivatives of the velocity potentials on the x-axis. Accordingly, regarding ~1
and ~ 2 as solutions of the Neumann problem for the lower and upper half-planes,
it is possible to represent them in tiie form of the corresponding potentials of a
simple layer and to check to see that the dependence
v ~2 a~ ~
_ ~ C~' + ~Z ~ a P ~ P~ ax p~ PZ ~Z ) .
,
- exists between them on the x-axis. Using this additional expression, following
from the continuity of motion, the boundary conditions for the velocity potentials
are transformed to a form similar to the Cauchy-Poisson condition at the free sur-
f ace, ~ _
_ d 1~, a~~, d~~ p, d_
Z~t. t.P a...~. o,
at= +~e aX2 ar at~. aXZ . f ay
where ~
_ ~S P, - Q~ 9 , .P,P~ C~~'~2)1 , ' ,
p,tp2 9CP,2-Pi~ .
For determining the ordinate of the discontinuity we also derive the simpler ~~r-
mula � . ~
~ ay,+.,p2~c,-c) dy: ~ (s)
% C Ot + ~x J r�o
, . . P~ P~ .
_ Second, in this problem the initial values of the functions ~ 1, ~ 2 and ~ have a
physical sense. However, the difference in the kinematic conditions (3)-(4) shows
that at the initi~l moment a zero velocity should correspond to a zero displace-
ment of the discontinuity, and on the contrary, a zero displacement corresponds to
equal velocities of the particles. This means that it is possible to stipulate only
two functions arbitrarily, for example,
- [ \X~ 0)a~\x/~ pJ 7J\x~Di ~)~)I ~Z `~~Oi ~~_r `X/~ ~6~
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where F(x) characterizes the difference in the impulse pressures imparted to the
boundary particles of fluid flows. Using conditions (6), it is possible to find
the initial values of the functions ~1, ~2, ~ and their first time derivatives
(2]. In particular, the second initial condition for the function has the form
a? (x : _ ; N.
a_______ = p~+ PZ F ~X~ ,
where F'(x) is the Gilbert transform of the derivative of the F(x) function.
The separation of the boundary and initial conditions affords possibilities for
use or application of the methods of the theory of functions of a complex vari-
able. On the basis of formula (5) using complex potentials it is possible to in-
troduce the function s(z;t), analytical with respect to z= x+ iy in the lower
half-plane, whose real part with y= 0 coincides with 1~ and obtain for it the
- equation -
�'IZtie=+`~as�0 c>>
at aZ
and the initial conditions
o~, f~X~ + ~
f~~~ , as ~x. o _ . [ F'cX~ - ~ F'~~] .
_ s ~ P~+ PI
3. We will examine the following solution of equation (7):
- i Z l~ / z
E~Z;t~,_ ~ e 1s ye z~~lt 1.
. ll ~
Taking into account the atnbiguity of the Hankel function of the first kind enter-
ing into it the real E1 and im~.ginary E2 parts of the limitin~ value of this
solution on the real axis from the direction of the lower half-plane are repre-
sented by the formulas ~
EiC,t; t~' I~% ~N, ~P) -~tP)s~ I~'
~
Ej(x:~)--1' . [a,(p)s~l~i?~CP~s ~ ~uyzX.
~ .
Here ~~(P) is the Bessel function; N~(P) is the Neumar.n f unction;
p~ ~2 ~ z ; ~ ~ t- t ~ .
~,P
_ The general solution of the problem of internal waves is written in the form of
th.e convolution �
~ _ aE Cx - t f~x~ _ a ~ X ~ t * f
~x~ +
dt
+ E~(x;~'~ f~(X) +E (x:t) * p
= f~(x) � (8)
P~ PZ '
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4. We will investigate the type of internal waves caused hy the concentrated ini-
tirit displacement of the discontinuity of the flows at the point x= 0 with the
total area Q, that is, with f(x) = Q b(x), F(x) = 0. In this case from formula (8)
we obtain - , _ _ . _
a ~
~ E~~x; t~- Er ~x: t~ ~ .
~ at
~
`1'h~ Gilbert transform E2 in the x-c~ordinate consists of two parts: the function
E(x;t)2, taken with a minus sign, and a nonwave addition, who~e t derivative is of
t~e order of Z/P2. Accordingly, with large P values we obtain �
~ ~ Qr cos(,o- - . (9)
z.~ eP~~2 y
We will study the wave movement described by formula (9). As a result of the sym-
metry of the rise r(relative to the y-axis it is possible to limit ourselves to
- . .
the case when x~ 0. 2 .
QZ
0 z
-0,1 �
0,2
0 2p 4,p
- 0,2 .
Fig. 1.
The general nature of the waves is schematically represented in Fig. 1, which is
a graph of movement of the discontinuity of the flows at the fixed point x and
the instantaneous pattern of waves at some moment in time t. As unity for the ver-
tical scale we adopt the value Q/ 21T'e, and the parameters and t are equal to
~ ~ i /J .9iS ~ s ~7' ~1 �
4 1~ 4 '
We will examine the phase change B- P-~, caused by transition alox~g the x-axis
from a particular wave to the next, that is, Q Q= 2 n'. Carrying out differentia-
_ tion for x and introducing the notation - ~ x= ~1, we obtain a general expression
for the wave length ~ as a function of ~
p .
a = P � ; ~lo~
Formula (10) shows that with adequately large'C the origin of coordinates generates
waves of the length
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a= y~te = 4~ 2~-r ~c, - c2~ ,
9 P, p= c~~>
which, with advance along the x-axis are distended and at the points correspc,nding
to sufficiently large E their length becomes almost the same (81t x2/9t2) as fc~!'
waves at the free surface of a deep fluid [3].
Similarly, a general expression is found for the wave period T as a function of "C
_ -
r' 1~- -ye p
% T ~12~
With 2�~ the internal waves have periods of about 41i"x/jt, which are J~ 1+/~ 2/
l~l - P 2 times greater than the period of the waves at the surface of the deep
fluid. With the course of time the period T decreases and when t>>~ becomes close
to its limiting value
r= 1~- ye = 4~ ~~j,~. I c, - c2.~ . ~~3 ~
~ ;i 9 P~- PZ
This means that upon the elapsing of an adequately long time interval the fluid
level in this place will experience almust periodic oscillations.
, The formula for the phase velocity
. T.. _
, ~9~= ~P P .
shows that the waves move with a constant acceleration equal to j/2 B, which is
P1 -~O 2/ P1 + p2 times less than the acceleration of waves at the free surface.
In addition, the phase velocity of the shortest waves (11) is equal to
G = ~~a " + j ~~~-G2~ � ~14~
~ p~ pZ
The group velocity of the internal waves, like the waves at the surface of a deep ~
fluid, is equal to x/t.
Thus, in a coordinate system moving in the plane of movement in a horizontal direc-
tion with the mean mass velocity of the floes (1) and related to the x, y axes,
series of waves of the length 4Ji'Q are propagated from the origin of coordinates
in both directions; these waves mov$ to infinity, being distended in length and
accelerating. As time passes the distension and the acceleration of the waves at-
tenuates greatly, as a result of which a considerable part of the discontinuity is
covered with waves whose length, period and phase velocity are given approximately
by formulas (11), (13) and (14); their numerical values are cited in the table.
In a fixed coordinate system these extremal characteristics of the waves are as
follows:
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`Z ~ CZ C
~ _ 7l f' ~ ~ 'f' I'*' r
~Y ,re ~ (15)
~.r- r// ft C`a ~~J~ ~~=~,f~+CI-
. ~ i ~ ` ~
1,. /�a ~I, `
C
Table 1
l~2/ /~l cl - c2 m/sec
0.1 1 10
Maximum length of internal waves, m
0.001 0.0001 0.001 0.13
o.ol 0.0~01 0.01 1.28
0.1 0.001 0.13 12.94
0.9 0.06 6.07 606.78
0.99 0.64 63.73 6372.69
0.999 G.4 640.17 64016.74
Minimum period of internal waves, sec
0.001 0,004 0.04 0.4
0.01 U.013 0.13 1.29
0.1 ~.045 0.45 4.8
0.9 1.22 12.15 121.52
0.99 12.74 127.46 1274.55
0.999 128.03 1280.33 12803.35
- Minimum phase velocity of propagation of internal waves, m/sEc
0.001 0.003 0.032 0.32
0.01 0.01 0.099 0.99
0.1 0.029 0.287 2.87
0.9 0.05 0.499 4.99
0.99 0.05 0.5 5
0.999 O.O:i 0.5 5
~ We note in conclusion that formulas (13) and (15) for the wave period were derived
by L. N. Sretenskiy [1] from a special solution of the problem obtained in the form
of a Fourier integral and evaluated using the stationary phase method.
BIBLIOGRAPHY
l. Sretenskiy, L. N., "Cauchy-Poisson Problem for the Discontinuity of ~ao Flow-
ing Currents," IZV. AN SSSR: SER. GEOFIZ. (News of the USSR Academy of Sci-
- ences: Geophysical Series), No 6, pp 505-513, 1955.
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2. Nov~k, A. A., "Formulation of the Problem of ~daves at the Discontinuity of ~ao
Infinite Currents of an Incompressible Fluid," Manuscript deposited at the All-
Union Institute of Scientific and Technical Information, 3 October 1979, No
3457-79.
3. Stoker, Dzh., VOLNY NA VODE (Waves on ~dater), Moscow, IL, 1959, 618 pages.
~ ~
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IidVESTIG.'~TION OF EFFECT OF VERTICAL DENSITY STRUCTURE ON INTERNAL WAVES
Sevastopol' TEORETICHESKIYE I EKSPERIMENTAL'NYYE ISSLEDOVANIYA POVERKHNOSTNYKH I
VNUTRENNIKH VOLN in Russian 1980 (manuscript received 25 Apr 80) pp 143-150
[Article by S. P4. Khart3.yev and L. V. Cherkesov]
[Text] Abstract: The effect of vertical density
structure on the kinematic characteristics
of free internal gravitational waves is in-
vestigated. The investigations are made both
for averaged density distributions and for
models with a vertical fine structure.
At the present time a considerable percentage of the investigations devoted to the
fine vertical structure of the ocean are directed to a study of the role played
by internal waves during its formation [1, 2, 8]. Closely akin to these investiga-
tions are the problems related tu the influence of fine structure itself on the
field of internal waves [3, 4, 9]. Due to the experimental studies carried out in
these directions it was possible to obtain a considerable number of different esti-
mates of the size of microscale inhomogeneities having the nature of stratification.
Such estimates, in turn, m~de it possible to choose rhe models most completely tak-
~ ing into account the real density distribution in the ocean [3] and making it pos-
sible to analyze the distorting effect of fine structure on internal waves [4, 9].
In this article, in contrast to [3, 4, 9], we study the dependence of the elements
of internal waves on fine structure parameters. In comparison with [3, 4], we give
- a more detailed analysis of the influence of characteristic vertical scales of the
- "fine stratification" ("sheets" with a thickness from 10 cm to 3 m were taken into
account) and high local values of the V~aisala-Brunt frequency on the kinematic char-
acteristics of internal waves.
We will examine a plane layer of 0~
~
. o
:
~
10
t ~ �
! cycles/houriY! N~ xx
y.
14� cycles/hour I0~ F~
~ Fig. 2. Fig. 3.
u, n u
However, if a N/a z increases with an increase in the Vaisala-Brunt frequency,
the theoretical curve in the case of large Vaisala Brunt frequencies exceeds the
experimental data. The presence of currents with a vertical velocit shear will
change the energy influx to turbulence, in particular, with a ZU/ a z~< 0 the rate
of influx of kinetic energy will be less than was determined in accordance with
(21).
Thus, the derived dependences of the rate of influx of kinetic energy on the condi-
tions of density stratification are confirmed by the cited experimental data.
BIBLIOGRAPHY
l. Dylcman, V. Z., Yefremov, 0. I. and Kiseleva, 0. A., "Investigations of the Fine
Vertical Structure of the Temperature Field in Eddy Formations of a Synoptic
_ Scale," in press.
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2. Dykman, V. Z. and Kiseleva, 0. A., "Interrelationship of the Fine Structure
of Internal Waves and Small-Scale Turbulence," in press.
3. Ozmidov, R. V., GORIZONTAL'NAYA TURBULENTNOST' I TURBULENTNYY OBMEN B OKEANE
(Horizontal Turbulence and Turbulent Excfiange in the Ocean), tioscow, "Nauka,"
1968, 105 pages.
n
4. Korotayev, G. K., Spectral Energy Density of Wave Packets," MORSKIYE GIDROFIZ-
ICHESKIYE ISSLEDOVANIYA (Marine Hydrophysical Investigations), No 2, Sevasto-
pol', pp 41-47, 1977.
5. Phillips, G., DINAMIKA VERKHNEGO SLOYA OKEANA (Dynamics of the Upper Layer of
the Ocean), Moscow, "Mir," 1969, 139 pages.
6. Garrett, C~ J. R. and Munk, W. H., "Space Time Scales of Internal Waves," JGR,
Vol 80, No 3, pp 291-297, 1975.
- 7. Hayes, S. P., Joyce, T. ri. and Millard, R. C., "tieasurement of Vertical Fine
Structure in the Sargasso Sea," JGR, Vol 80, No 3, pp 314-319, 1975.
8. Brethertone, F. P. and Garrett, C. J. R., "Wavetrains in Homogeneous Moving
Media," PROC. ROY. SOC., Ser A, 302, No 1471, pp 529-554, 1968.
,
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A EFFECT OF SURFACE FILM ON NATURAL OSCILLATIONS OF FREE BOUNDARY OF FLUID
Sevastopol' TEORETICHESKIYE I EKSPERIMENTAL'NYYE ISSLEDOVANIYA POVERKHNOSTNYKIi I
VNUTRENNIKH VOLN in Russian 1980 (manuscript received 19 Jun 80) pp 175-181
[Article by T. M. Pogorelova]
[Text] Abstract: The effect of a film on the at-
tenuation of surface waves is investigated.
The dispersion law was formulated: a pre-
cise solution of the frequenc.y equation is
found. For any values of the parameters of
the problem simple asymptotic representa-
tions are obtained for the dimensionless
complex frequency s as a function of the
wave number (and wave lengths L). The de-
pendence of the period of oscillations and
the logarithmic decrement of attenuation
on wave length is found.
In many practical problems in geophysics it is desirable to investigate the in-
~ fluence of a surface film on the wave movement of a fluid. The studies of V. G.
Levich were devoted to this problem [2]. Iie examined the problem of the natural
oscillations of a fluid with a surface-active film taken into account. He found ap-
proximate solutions of the frequency equation under definite conditions superposed
on the parameters of the problem.
In addition, studies [1, 3, 4] were devoted to the problem of the natural oscil-
lations of the free boundary of a fluid without a film. In [1] the azalytical de-
pendence between the frequency of the free oscillations and the wave ~�ngth was
investigated numerically by the Graffe metho~ for a number of values of parameters
_ of the problem. An analytical dependence of the complex frequency on wave number
was found by E. N, Potetyunko in [4].
In this article we give an analysis of influence of the film on the logarithmic
decrement of attenuation and the wave period for the problem of the natural os-
cillations of a semi-infinite fluid covered with a film. The influence of the `
film is "conveyed" into the boundary coridition of the problem.
1. Formulation of problem. We will examine free oscillations, periodic along the
Ox-axis, for an infinitely deep two-layer viscous fluid occupying the lower half-
space,
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a~~ ~ vp, + ~;a i; +y~, d~~ i; _~0, p, _P, +p,9(~-~) ,
at ~ (1..1>
- a~ _ _ ~ P~ + y1a ? +9,~ , d~v ~ = 0, p ~ P _p yz.+p9fi;
at p 1
with z = 0 2
1
~ -1 +.Pc;? 2~r1 ~ZZ-~ ~ ~-~y.P,9~+1.r~~azj ~ c~.z~
; ~o~z .~y~~ a~;? ~~X.
~:~~ax + az ~ _ ~ ax + az ~ ,
_
l Z- I, X e X, d~1 /d~ -?z ,
_ ~zX+aXz-0, z, z=h; z---~, 1--D, P-0;
-p+o ayj-~l a?~' 0, z=h. c~.3>
~~,9~ p9 .~,~z
Here OC1, 4~2 are the coefficients of surface tension at the upper and lower dis-
continuities respectively; /~1, /~2 are the densities of the upper and lower flu-
- id; ?2 are the coefficients of the kinematic viscosities of the upper and
lower ~luids; ~v 1, Tj2 are the disturbances of the free surface and discontinuity.
The origin of coordinates is taken on the undisturbed discontinuity of the fluids,
the Oz-axis is directed vertically upward; h is the thickness of the upper layer
(h-~ 0) .
2. In place of precise satisfaction of the equations of motion of the upper layer
from (l.l) we require their satisfaction in an integral sense: we will integrate
the equation~ of motion of the upper layer in the thickness h with the boundary
conditions (1.2) and (1.3).
As a result, the initial problem is reduced to the following boundary-value prob-
lem: _ - ~ O'i0 ~Vx
2
dt ox ~'i Il)~
.P?
cz.~>
av~= _ ~ a~ + ~ (o~VZ+ ~v l, a~ + ay = o ,
dt ~ ~ldx ~l ~
,P dx dz
the boundary conditions with z= 0 are
? d~/
_p + ~ a~= a ;?~p, ~ ~ ?x ~ D~
1 p 9 2 dz aX aX az c2. 2>
~~2 ~ i ; ~ ~ , ~ D.
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Here OL~ = JC1 = aC2.
3. We will seek solution of system (2.1) in the form
1 ~F(z)e~~Xedf z=.?(z)e`~e~f pa~~I~e~gXe6t, (3.i)
where a is the frequency of the natural oscillations; is the wave number ($7 =
2~/L; L is the wave length).
Suhstituting the solution (3.1) into system (2.1), we find
~ e6z_ e~Zl e~fx ~ 6t ~
C ~
~ p c3.z>
? _~~Zear_ de le ,
~'Z ~~X, e6t P= c~ e~I e`~X e dt
,p2 /
~ Here t~ +t7/ ~J 2; Re B>0. Satisfying the boundary conditions (2.2), we obtain
the following system relating the unknown constants cl and c2:
-c a~62+9~+Z26~2+~0 +c2p [g+2Z6~ +,~o~'Zl = 0,
. l-2C~~'~+~ 61 ~2~2+ ~/~1~ (3.3)
Excluding (3.3) in system cl and c2, we obtain the frequency equation
-6Z f.~o~~- f9 t~?2~'~ ~1+6/~2-~~4vZ-`~`~?26=0. c3.4>
Solving this equation we f ind the dependence of 6 on the wave number .
We will carry out a replacement of the variables in (3.4)
- 6= 5~1~1 , (3.5)
where s is dimensionless frequency. Then equation (3.4) assumes the form
(61+~~2-~6 + (3.6)
Here 1
~ +9 ~ +~l
~ ~~1 . 6= ~+s , ,l�o-= ~ ~3.>>
~
4. The roots of the dispers3on equation (3.6) are algebraic functions of the para-
meter T and are represented in the broadened plane " 8" as a four-v~lued f uncrion
of 1 I~?~ 5l . . . _
a~ ~I -,to)k~r ~ o~ 6
k lu k=su
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Substituting (4.1) into (3.6), c~e obtain the following representation for the
complex frequency s:
S= C .~~C~-~_ ~ (4.2)
�
?=0
' ~ ' n-1
c~ =`o [~a-.~' ' - ~ ~k z_k , ~a5~
1 k_1 J
CO=QZ, C=O, ~ ~ a?/QOr ~'4,~oi
~2
Q~ 1Q a~a-k~ ~ n35,
~
a 'V` l~*~~/~, Q~=~~ Qa- ~?d0, Q'~=~~?, Q'4=-~8~ (4.3)
0
o~e
,,1 _ ~!'
E -t ~ C a ~"+~aka~_k , ~32, ~
.J ~ ' ~ k=9
o=B1, c_8~/28, B=6B*?~-2. 8,~~8~,
. . ~
1
Qa'� _ ' 8 ~ a~ Q _k..~ +B ~ d Qn-k ? ~ a~F ~n-~ fJ ~ ~a3,
. ?8a, . k,, k.,
k
~ ~ ~ B~ k :
a,~~.,. Qs-~~, d~~a,~ak_~,;,,
- nt�! (4.4)
, S - ~ ~ -
~ a=0
~
_ ~ C? = Qk Q k, R Co = Q
Z. c7a =
- and the second value a~ satisfies the equation
o~+Qo +,~Qo 0,
Q?~- ~ f.Z/~~Q~,Q~,+~Ck ~_kl , R~Z,
~L ~ k�! J
~
Q~ ~ - ~~4/~, /!l~ ~ Q
1 + / , LZo Qo +
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~ ~
' 46~ - Ca, �
~
B* is a positive root of the equation
~t~~~-'~0.
5. Limiting ourselves to the first terms in formulas (4.2)-(4.4), we have the fol-
lowing simple representations for the complex frequency s:
_ ;
0~.~>
1 becomes equal to zero. In the range of waves with a length of the order of sev-
eral depths, regardless of the relationshi~, ;~etween AL and AZ, the approximation
(14) seems quite good; the short-wave and long-wave approximations (15) give some-
what exag~erated results.
In connection with the above, it is desirable to examine the second approximation
of the expansion in the small parameter F,. For short waves
6?- 2w~
.ri'r
6 -4w
The r4 and r5 values, regardless of the length of the surface waves, exert no ap-
preciable influence on the value of the decrement of attenuation.
Thus, the first approximation for E1 and the second for ~2 give a virtually pre-
cise solution of the formu].ated problem.
BIBLIOGRAPHY
- 1. Vaz~v, V., ASIMPTOTICHESKIYE RAZLOZHENIYA RESIiENIY OBYKNOVENNYKH DIFFERENTS-
IAL'NYKH URAVNENIY (Asymptotic Expansions of Solutions of Ordinary Differen-
tial Equations), Moscow, "Mir," 1968, p 172.
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2. Goncharov, V. V., "Some CYiaracteristics of Internal Waves in the Ocean,"
TSIT:IAMI I VNUTRENNIYE VOI;NY (Tsunamis and Internal Waves), Sevastopol~, 7zd.
MGI AN UkSSR, pp 87-96, 1976.
3. Yerugin, N. P., KNIGA DLYA CHTENIYA PO OBSHCHEMU KURSU DIFFERENTSIAL'NYKH
URAVNENIY (Book for Reading in the General Course on Differential Equations),
t-iinsk, "Nauka i Tekhnika," 1972, 660 pages.
4. Zadorzhnyy, A. I., Attenuation of Long Waves in an Exponentially Stratified
Sea," MORSKIYE GIDROFIZICHESKIYE ISSLEDOVANIYA (Marine Hydr.ophysical Invest-
igd~~~ns), No 3, Sevastopol', pp 96-110, 1975.
5. Levkov, N. P., "Dissipation of Surface Waves Generated by Periodic Atmospheric
Disturbances," MORSKIYE GIDROFIZICHESKIYE ISSLEDOVANIYA, No 4, Sevastopol',
pp 76-83, 1973.
6. Moiseyev, N. N., ASIMPTOTICHESKIYE METODY NELINEYNOY MEKHANIKA (Asymptotic
Methods of Nonlinear Mechanics), Moscow, "Nauka," 1969, 800 pages.
7. Nerkesov, L. V., GIDRODINAMIKA POV~RKHNOSTNYKA VNUTRENNIKH VOLN (Hydrodynam-
ics of Surface and Internal ~Javes), Kiev, "Naukova Dumka," 1976, 364 pages.
8. Dore, B. D., "The Decay of Oscillations of a Nonhomogeneous I~luid Within a
Container," PROC. OF THE CAMBRIDGE PHIL. SOCIETY, Vol 65, No 1, pp 301-307,
1969.
9. Fliegel, M. and Hunkins, K., Internal Wave Dispersion Calculated Using the
= Thomson-Haskell Method," J. PHYS. OCEAN., Vol 5, No 3, pp 541-548, 1975.
10. Hyun, J. M., "Internal Wave Dispersion in Deep Ocean Calculated by Means of
~ao-Variable Expansion Techniques," J. OCEAN. SOCIETY JAPAN, Vol 3?, No l,
pp 11-20, 1976.
11. Johns, B. and Cross, M., "The Deca~ of Internal ~Jave Modes in a Multilayered
System," DEEP SEA RES., Vol 16, No 2, pp 185-195, 1969.
COPYRIGHT: Morskoy gidrofizicheslciy institut AN USSR (tiGl AIJ USSR), 1980
, 5303 - END -
CSO: 8144/1944
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