SCIENTIFIC ABSTRACT ALEKSEYEV, A. S. - ALEKSEYEV, A.V.

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• Collection: Scientific Abstracts
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• Publication Date: December 31, 1967
• Content Type: SCIENCEAB

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iiaSE-YEV, A.-S.I.-GELCHIISKIY, B. YA.------ -- -
"Successive Approximations in the Ray Theory and its Applicstion in Propzigation
Problems Involving Boundaries."
"Ray Theory of Intensity and Shape of Leading Waves in an Elastic Medium."
paperSpresented at the 4th All- Union Acoustics Conf., 26 May - It June 1958.

,4 k- _F V j 49-1-2/16
AUTBDRS:Babich, r.M~. and Alekseyev, A.S'0 __
TITLE: On the Ray Method of Calculating the Intensity of Wa-ve-
fronts (0 luchevom metode vychisleniya intensivnosti
volnovykh frontov)
PERIODICAL: Izvestiya Akademii NaWc SSSR, Seriya Geofizicheskaya,
1958t Nr 19 pp'..17-31 (USSR)
ABSTRACT: The growth of dynamic seismology leads to the necessity
of calculating the intensity of longitudinal and trans-
verse waves in inhomogeneous media at the reflection of
the waves from curvilinear boundaries. Such calculations
can be carried out by considering the relations obtaining
on the characteristic manifolds of the equations describ-
ing the wave processes. Analogous considerations lie at
the basis of the methods of Hadamard (Ref'.1) and Sobolev
(Ref-.2) for the solution of the Cauchy problem for hyper-
bolic equations. The method described in this paper has
previously been applied to Maxwell's equations (Ilefs".3-5)
and to the wave equation (Refs'.6-9). Levinand Rytov
(Ref'.10), and Zvolinskiy and Skuridin (Refs.11 and 12)
have applied ray considerations to the equations of the
Card 1/9

49-1-2/16
On the Ray Method of Calculating the Intensity of Wavefronts.
theory of elasticityp but in none of these,papers are to
be found the equations 4'.2l 4".3, 4".5 and 4.7, which are
at the basis of the method described. The method of des-
cribing the function f(cx , a ) for a concentrated source,
which is an important pari of2the method, is also new.
Let t = 't (x, y, a) be the equation of the wavefront at
time t . Let the wave process under consideration be
described by the scalar or vector function U(xj yp z, t)
where it is assumed that
U(XVYVZI,t) ~ UO(XOYPZ)fO(t - 'V) + Ul(xtytz)fi(t
+0 (f 2(t
in which
f I (t) = fl (t) f 1 (t) = fo (t)
2
It is assumed that in some sense the function f 2(t) can
be neglected in comparison with its derivative. If Eq~*(Jrl)
is substituted into 1
U + U U 0
Card 2/9 xx yy C2(x, Y) tt

49-1-2/16
On the Ray Method of Calculating the Intensity of Wavefronts.
and the coefficient of fol equated to zeror there results
2 grad V grad U0 + U0AV = 0
which is studied in some detail. Equations analogous to
Eq' (2'
.4) for the case of an inhomogeneous elastic medium
derived by substituting the expression for the vector
U(x# yj t) from Eq!.(f.1) into the two-dimensional differ-
7;ntial.equations of motion of an inhomogeneous elastic
medium'. Thus we have
)9rad J1U (grad *r)2 + 0
- (X + p) (grad tU U = 0 (Bq'. 2.6)
-0 /-0 1-0
(7, + )a) (grad tU:L)grad -C pUl (grad t)2 + PUl = 0
(Eq.2'.7)
where
Card 3/9

49-1-2/16--
On the Ray Method of Calculating the Intensity of Wavefronts.
g(U0, %) = (7, + A) ~(div Uo)grad '[ + grad (gograd -Q] +
+ U NJ AT + 2 (grad J/O:,grad T)'i + 2 (grad Uoy grad T) +
0 , i1
+ grad X (Uo grad Z) + (gradul2o)grad 'C + (gradu grad t)]go
(Eq.2.8)
and U , U i, j are unit vectors in the
Uo = t ox oyl
directions of x and y respectively. Eq.(2.6.) is a
system of two homogeneous equations in the two unknowns
U ox and Uoy 1 and it can be shown that the determinant
of this system only vanishes in two cases. These are:
(a) when 1grad -r - 1 -P (longitudinal wave) in
f a7 - x + 2)11
which case we shall write 'ta for 'U ; and
(b) 1grad 'tl2 P
b
we shall write Tb for
Card 4/9 shown that:
(transverse wave) in which case
T. In the first case it can be

W-7.0
Wo
49-1-2/16
On the Ray Method of Calculating the Intensity of Wavefronts.
U = Yograd Za (Eq.3'.1)
..o where (Po is a scalar
function of position'. Eq.(2.7) can also be regarded as
an algebraic system of equations for the unknown compon-
ents of the vector "g~j . Again there are two conditions
for solution, the first of which can be written in the
form :
I, (U 0, Ta)grad Ta= 0 (1Sq .3 .3
If Eq.(3-.l) is substituted into Eq.(3-3), after some
simplification
2 ~Po + [a 2A-t - (X + 2)a) rad I grad Zalgo = 0 (Eq-3.6)
~It;a a (9 P
is obtained. In Fq.(3~.6) the derivative is calculated
Card 5/9 along the ray of the longitudinal wave. If Un and U 1)
P-P

49-1-2/16
On the Ray Method of Calculating the Intensity of Wavefronts.
are the components Of the U 0 along the normal and the
binormal to a ray of the transverse wave, then the condit-
ion for the solubility of the system 2.7 can be written in
the f orm:
j
2 -16 u n+ 9 Thu + (b2gr' + gradu grad Un = 0
--D "b
BTb P
DU9 2AU 1 (3.8)
2 2 ThUn + (b + 7grad)a grad T u = 0
altb b, b) 'i
Suppose that a point on the ray is characterized by the
quantity and the ray itself by the parametex a
and let x x(a, T Y = y(m, Z or, in vector'form,
X X (a,T Eq.(2.4) can be written in the form N
2 3UO U,
+ 0 ict 0
Card 6/9 07 Tr (ILL)

On t%e Ray Method of Calculatix)g the
IU,, I f (P)
pb
In the three-dimenBional case
the two parameters a 1 and
have their analogies in:
49-1-2/16
Intensity of Wavefronts.
(Eq~.4~03)
a ray is characterized by
m 2 and Eqs.(4.9-) and (4'.3)
IU01 - 1 =_ f (al ' 'T2 (Eq'. 4'*5
Vja?a
and: luol f (0-1f 02) (Eq'.4".7)
"fjb Pb
where J The authors conclude by consider-
ing three examples: (1),The reflection of waves from a
curvilinear boundary; 2~ Media whose inhomogeneity de-
Card 8/9 pends on 1 coordinate; 3) The diffraction of a cylindri-

49-1-2/16
On the Ray Method of Cale-alating the Intensity of WavefrontB.
cal wave at a wedge
There are 3 figures and 28 references, 21 of which are
Slavic-
ASSOCIATION: Ac'. of Sciences of the USSR9 Leningrad Branch of the
Mathematical Institute imeni V.A!.Steklov (Akademiya nauk
SSSR?,- L'pningradskoye otdeleniye Matematicheskogo instituta
im. V.A. Steklova)
SUBMITTED: July 29, 1956.
AWLABLE: Library of Congress.
Card 9/9

AUTHORS: Alekseyev, A. S., Gellchinskiy, B. Ya. 2o-3a8.4- 1 o/61
-----------------
--

TITLE: -----
On the Determination of Head Wave Intensity by the Method
of Rays in the Theory of Elasticity (Ob opredelenii
intensivnosti golovnykh voln v teorii uprugosti luchevym
metodom)
PERIODICAL: Doklady Akademii Vauk SSSR, 1958, Vol- 118, Nr 4,
pp. 661-664 (USSR)
ABSTRACT: The present paper investigates the intensity and the shape
of the head waves occurring at the planar boundary of elastic
media in the case of a linearly polarized wave with an
arbitrarily shaped head wave striking this boundary. In
rectangular coordinates x,y,z let z> 0 and z 0 containing the elastic
medium, f, denoting the density. The elastic properties of
the medium taking up the half-space z 0 in cylindrical coordinates rt^,%
z. Lame* parameter X is an arbitrary, function of r,13; z; ~L and p
depend on z only. The half-space is filled with an elastic medium
at rest. At time t - 0 a surface moment
(5z . 0, 'Crz 7 Of T,~z w a(t)b(r) when z - 0
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S/049/62/000/011/002/006
Some converse problems D218/D308
is applied to the surface z - 0. This gives rise to SH waves only,
and the elastic displacement vector is
U (r, z, A) U (r, '&I Z.1 t)
where U-CA is the component of the displacement vector in the
direction of the unit vector'k, . The r and z components
are Ideritically zero. The differential equation for UO. is
2 2U'd ~L, (z)C)U,
U,& U P(z) 82 UA,%
br2 r .'0 r r2 Z2 (z) bz 11 (z) j3 t2
(3)
subject to the boundary condition
a(t) b (r) (4)
6 z z 0 (0)
Card 2/4

B/04 62/000/011/002/006
Some converse problems ... D218YD308
and the initial condition
Uft I t < 0 0
The development of these equations is considered in greater detail
in a previous paper (Doklady.AN 333R9 v. 103, no. 6, 1955)- It is
assumed that a(t) a 6(t) (Dirac delta function) and
CO
1 d 6 W
b(r) k2jj (kr)dk . - r
Tit- 177 W-r r
0
where J, is the first-order Bessel function. It is shown that the
problem may be reduced to a standard Sturm-Liouville boundary value
.problem and that the final solution may be written in the form
CID
U (r, Z, t) (x, t, k)k2il(kr)dk
0 (36)
Card 3/4

-001, M!
5/049/62/000/011/002/006
Some converse problems ... D218/D308
C1 (X It I k)' CD a i n 1-5, T(xtj )d
S Yk Pk
0
z
X (K )) d~ 1 6 (z) -~ji (z)
0 (37)
where is the Lam6 parameter.'This solution is used in the
following paper (Izv. AN 83SR. S. geofiz., 1962, no. 11, 1523 1531)
to formulate the converse problem.
ASSOCIATIONt Matematicheskiy institut im. V.A. Steklova,
Leningradakoye otdeleniye (Ma:thematical
Institute im. V.A. Steklovo Leningrad Division)
SUBMITTEDs April 20, 1962
Card,4/4

43344
s/o4g/62/000/011/003/006
D218/D308
AUTHORt Alekseyev, A.S.
TITLE: Some converse problems in the theory of wave.
propagation. II. The spatial problem for SH
waves (the converse problem in an overdeter-
mined formulation) J/
PERIODICAL: Akademiya nauk SSSR.'Izvestiya. Seriya geo-
fizichaskaya, no. li, 1962, 1523 - 1531
TEXT: This paper is a continuation of the paper on
PP- 1514-1522 of this journal..It is requIrea to determine.AL(Z)
and f(z) in such a way that '
U~,(r, 0, t) 0 (r, t) (4)
where G~r, t) is a function known in the range 0 < t -- T, 0 -4 r iC- R
and _Uj, rp 01 t) is a special solution of the problem defined by
(3) (5) of the preceding paper. The function G(r,t) is assumed
to satisfy the following conditionst (1) there exists a curve
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s/o4 62/000/011/003,/006
Some converse pro*blems ... D218YI)308
~ r~(t) on the plane (rot) such thai UO(r, 0, t) - 0 for
~ ;r1 t); (2) the function PkW vanishes for X < 0 and does
not decrease with increasing ON for any k in the range (0, co);
if the condition
C5 2 Vf -X where > 0 (11)
k P k
is satisfied, then the function
OD Cos x
a - (X) S dd (12)
k k
has continuous derivatives up to the fourth order inclusive for
any k ) 0; and (4) in the range 0,~ t 4 T the function t (0.0,k)
has three continuous derivatives with respect to t for any kbO
and may be written in the form
(0, t, k)* 00 1 Co. -4 kt d W
S x Pk
0
Card 2/3

S/049/62/000/011/003/006
Some converse problems ... D218/D308
where W (k) is a function which does not decrease %!ith increasing
>,. Subject to these restrictions, it is shown that there exist
doubly differentiable functions p(z) and P (z) which in a certain
interval CO, Z(T)l define the medium uniquely'for the above for-m
of G(rit). The minimum conditions for the problem will be cone idered
in a future paper.
ASSOCIATION: Matematicheskiy institu t im. V.A, Steklova,
Leningradskoye otdeleniye (Mathematical
Institute im V.A. Steklovq Leningrad Division)
SUBMITTED: Aprii 20t 1962
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R
I AN S

s/141/62/005/002/018/025
E14o/E435.
AUTHOR: Alekseyev, A.S.
TITLE: The method of point transformations applied to the
cyclical:. production of an arbitrary number of
articles on a single equipment in the pre6cace of
varying backlogs
PERIODICAL: Izvestiya vysshikh uchebnykh zavedeniy. Radiofizika~.
v.5, no.2, 1962, 370-384
TEXT: The author presents algorithms based on the method of
point transformations (known from the theory of nonlinear automatic
control systems) for planning the production cycle in a system
where different articles are to be produced on a single machine
or group of machines or, rather, to select'among various possibl:e
such plans an optimal one, The -system is considered to be a
nonlinear dynamic,system with continuous parameters, the
differential equations are s~t up and stable limit cycles,sought.
These limit cycles must start from some known initial conditions
and arrive at carrying the procesai~-cyclically__- through certain
Card 1/2

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circuits.
ASSOCIATiON: Nauchno-issledovateltakiy fiziko-tekhnicheskiy institut
pri Gorlkovskom universitete
(Scientific Research Physicotechnical institute at
Gor'kiy University)
SUBMITTEDt December 29, 1961
Card 3/3

VOLIVOVSKIY, I.S.; YERMILOVA, N.I.; KRAUKLIS, P.V.;
RYABOY, V.Z.
Physical nature of certain waves rec-orded in hodographic seismic
Sounding. Part 1. IzY. AN SSSR. Ser. geofiz. no.lltl620-1630 N
163. (MIRA 16:12)
1. Kontora "Spetsgeofizika", Leningradskoye otdeleniye
Matematicheskogo.instituta imeni Steklova AN SSSR.

IGOLITSMAN . Fedor Markovich; ALEKSEYEV, A.S.; nauchn. red.
(Principles of the theory of interference reception of
regular waves) Osnovy teorii interferentsionnogo priema
reguliarnykh voln. Moskva, Nauka, 1964. 283 p.
(MIRA 17:12)

ACCMION NR: AP4014023'--
AMHORS% Aleks!7!Tv.A. So; Vollvovskiyo 1. So; Termilovap No L; Kkauklist P. Vol
T&bCq. V~-,'7o,_-_--_--
F
TITLEt The physical nature of some waves recorded during deep seismic sounding*
2# Theoretical analysis of models of the earth's orust for regiono of 0#4#4 A*4
SOMORt AN BSSR. Izv. Seriya, geofixicheekayap no. ip i964t 3-19
TOPIC TAGSt deep seismic sounding, earth's orustq Central Asia, head wave,
refleoted wavep refracted wavel kinematic characteristic, dynamic charaoteristiop
Turkmeniap shot point, apparent wave velocity
0
AlbMLCTs The authors present results on theoretical comparisons of the kinemati
and, dynamic characteristics of the earth's crust in southeastern Turkmenia. They
have considered possible laws governing changes in apparent wave velocity with
.-!distance from shot point in layered inhozogeneous media with plane-parallol inter-,
faces. Three different models of the earth's crust were ueedq based on different
valuesp densities# rates of change with depthp and ooabinationo of theseo-,
k,-Iftsults show that in layeredp inhomogeneous media the following relat a 8,
;~Jhold for the different k4nd of waveeg for head waves dV*/dx - 0 and d V* Ov,
Card 14yu?'-
77

ACCESSION NRt AP4014023
J..
2V*/dx2>0; and for refraQted waveop if
t7f, for refleotdeA waveR dV*/dx < 0 and d
d2j*/dx~, > 0 or 0~*/dx2 e
dV*/dx