SCIENTIFIC - PHYSICS, WAVE MECHANICS

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP80-00809A000600240292-1
• Release Decision: RIPPUB
• Original Classification: C
• Document Page Count: 18
• Document Creation Date: December 22, 2016
• Document Release Date: July 6, 2011
• Sequence Number: 292
• Publication Date: July 22, 1949
• Content Type: REPORT

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REPORT NO.
THIS IS UNEVALUATED INFORMATION
Zhurrnal Tek2micheskc Fiziki, vol SPITZ, No 4.
PMB -'er. ribs 5i T95 -- Iatoraation requested.)
PUBLISHED
DATE
PUBLISHED
CLASSIFJ TION.
CENIiAL INTELLIGENCE AGENCY REPORT
INFORMATION FROM
FOREIGN DOCUMENTS OR VADIO DROADCASTS . CD NC.
Scientific - Physics, wave mechanics
Monthly periodical
Leningrad
Apr 1948
t(E CF 1948
IffQRMATION
DATE DIST. 'r'"/.Tun 1949
NO. OF PAGES 18
SUPPLEMENT TO
LANGUAGE Russian
11111 POCUI1IIT corrAltt IIIPotnAT10N A7710TIN5 THE NATIOAAI. 1171X51
07 THE UNITES STATED WITHIN THE 01A1151 07 E4f1ONA61 ACT PO
W. S. 0.11 AND St. Ai A155151. ITS TURNS-1911ON OR TNt EITIIATIOA
SOURCE
R&FLSCTIOE OF SPHKRICAL WAVES FROM PLINK.
BO1 TARY F SSPARTiN11BTWF7 TW') M CIA
L. Brekhoveki'kh
Phys mat iced Sol US"
Submitted 17 November ]' 7
f Figures referred. to herein Fire appended.)
This article studies the reflection of spherical sound and electro-
magnetic eaves from a plane boundary of separation between two media. The
work of Seramrfeld [iJ, Weyl [2 7, Fok [3J, Leontovich [4J and others
on electromagnetic waves has been matnly connected with the cases of highly
reflecting boundaries corresponding to the condition
)6,1 > A
vhere8,is the dielectric constant of the ground. These conditions are fulfilled
when the conductivity of the ground is sufficiently great.
With regard to shorter radio waves, cases where Ej is small and Many
cases of propagation in the troposphere where e, is close to unity must be
studied.
STATE
ARMY
CLASSIFICATION
~
NAVY x NSRS
DISTRIBUTIO^I
LiA3 PSI
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In acoustics condition (1) is not satisfied in most practical cases, for
instance in sound reflection from the surface of ground or water or from the
bottom of the sea.
The problems of electromagnetic and sound waves have not been fully solved
up to the present time.
This article will restate in greater detail previously published results
of the author's research f 10-7 and show that the chief result obtained by Ott
4ro
~9 was inaccurate (bee end of S2). And, whereas results obtained by Krffger
L 8J J and Ott apply only to angles remoxe from the angle of full internal re-
flections, this article will describe the fields at all points of the area in the
wave zone.
The results shown facilitate clarification of many problems including oases
where `i1 In large or close to unity and establish a criterion for the application
of goamotrical optics (acousticp) in which a wave issuing from an imaginary
transmitter may be substituted for a relected wave.
I. 3TATW4 T OF TO FROBIBM -- FIRST FORD OF THE SOLUTION
Lot, us examine the point emitter at distance z0 from n plane boundary
of aeparnt:on between two media (Figure 1 For th, electromagnetic instance
let us select as transmitter a vertical dipole, for the acouatic instance, a
pulsating sphere with an infinitely small radius. After assigning the trans-
mitter frequency in a ectrudymmics, we shall denote the upper and lower media
by the wave numbers ko and kl, k, hating any imaginary frequency and ko a 9ma1?
calculated imaginary frequency.
In acoustics, in addition to the wave numbers, the media will be denoted by
the densities e. and e, .
We shall denote the field by the scalar function 191, which is the vertical
component of a Hertz vector in electromagnetice and the sound potential In acoustics.
Ths wall-known expreesir;i for a primary wave, sometimes called the initial
excitation, has the form (L 3], pages 941, 944):
,P? eR o 0 y rrl~r~e /~ ~e Fo Zp~, (2)
? ~ Ko 0
where the plus sign is used when n mount.
'II. G!f)M.gTRICAL OP'T'ICS AS A FIRST APPROXIMATION
In the simplest case of an.absolutely reflecting boundary, when
the reflected wave is reduce to form (3), except that a+ zo replaces z - zo in
the exponent. Hence we shall have
R
where R= 2 ? (z~zo) is the distance 'ram the observation point P to the
imaginary transmitter ql (L".aae 1). Here ai'o the wave reflecr_.inn v11i be
spherical. It has been generally supposed that this was true only in the case
of absolutely reflecting boundaries, but it is evident fret (4) that in all
cases where the reflection coefficient does not depend on the angle of incidence,
that is when f (or) = C, the reflected wave will be spherical
(9)
This case occurs. for example, in acoustics when cc = cl, (conditions apvroxi-
mating this may occur when sound is reflected from certain forms of sea .)ottoms),
when the reflection coefficient equals C Pt -'Po
e,+
t 1 T1AL
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4- 1
V;x e 0
The reflected wave will now be expressed in the form
r 1~k? a a,;not - r4_.~~~a Ciko ReosXa, that is, at angles at which a plane
wave does not have full inner reflection, the reflectod wave will be fully ex-
pressed by (17) (only If X-Xo is not small, see IV).
If we runs decreases angle X , contour r will shift to the left so that,
p.iint Sl will pas t';. wgh it and be on the right (Figure j). Now, if the
profile be disregarded, the beginning and and. of contour/- will lie on difr
ferent lobes and it will be impossible to cross over to it from contour / lying
entirely on the ul er lobe. In this case, let us take the integration path shown
in Figure 3. It goes from point J i..)(-into point-t ao , whore it meets the
profile and goat: along one margin cf it to S1, ord. than back along the other to
co . Then it intersects the profile and, already at the lower layer,
goes to point tco , and thaaoe along the normal path P , the beginning
of which now flee on, the lower lobe. Thus it a necessary to add to expression
(17) an integral like (il), but taken along the profile margin (contour r I.
Figure 3). Moreover the mall term in parentheses behind the integral
sign may be 4isa-rded. t o e cosot ,
It is convenient to divide the integral into two for the two parte of
the profile margin, whereupon the expressions behind the integral sign will be
distinguished only by the radical
q - V:1 LOS 2~
(21)
Reversing the direction of one of them, and combining both expressions,
we shall obtain for that part of a reflected wave 99a. (denoted by 9P,* ), which
is dependent on integration along the profile margin,
ii r- i IT
1Jir B ~(d)eakoRcar(X-0C) cos ads (22)
where
L
~~a~c_l9r4Sina2s
7A2 Sin,20- C7
(I7n q > o):
and the profile line is denoted by L.
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By introducing a new variable of integrations=a-',;the exponent unier
e 2 ko Rc,s (x- a'i . = e ilk lc [ors (-~c) t-w ;~-.r in(Xe-X) s i ~),
The right side of this equation can also be written
eika1A--1 -2rkoR~ (Xa?X)siy? rs;o(x,-X)sin .o 1. (25)
Be3ause kor is large, the. important expression behind the integral sign involves
only small f , of an order of magnitude not greater than
IN9114 " (26)
Hence, in converting the remainder of the expression behind the integral
Here we may discard 0 as compared with 2 469 which can always be done
when X, is large. When they are small, we have .1-- n and,, conasquently,
in accordance with (26), we must pustulate that
. =sin (a' X,) s'n ( --~(o)'~sin (2Xa?--P)sin q=sin P.
gcas17 a - ccs aareos;(,-?~os a)(oosa r o~aa)~
i k,R0--x,)1>i?
Hence the case where n is Very close to unity must be excluded from ou,? study.
After inking similar eliffimtions, expression (22y ma be presented in the
form
N B/2~'a'l,G xo..-ice 3
~r ~?/nrsin 7 e2k~RCaes( -r;n0( -X)si? (r~2
xo p 2sin
As demonstrated by V. A. Fok, the last integral may be expressed by t
Weber function ('12J (the function of a parabolic oy~}}inder)e For this It
is neoeseary to use the equation demonstrated by him CMteqP4 (29) may be
expressed by a Weber function without recourse to relation (30), If we tale
the function behind the integral sign in form (25) and substitute sin. = t,
and cos-- c/ , the integral obtained coincides with those forming part of
the integral representation of Weber's function (L 12J, page 157), But it is
stdficiant here and later to use formula (30)#)
7r -fen
2 a13 -
I'-n) a (s'-,~a)cosP-' /i tq 2i Z-,,4=l)
CAL
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The asymptotic expansion of x(p)ioas the form:
)_I_ 3 5 9 ? s I7
2-4
7.9?//?13+ 7.y.. !9.2/ _...1.
.q? f(q 2)z 4 .6.8.,O(8 )2
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z {Ro
our integra3i comes to this, if we put
?kaR cos xa -X, ~= 1k, Rs~n Y? >
qqr = 4 ~1. (32)
As inbteail of the ff un ti ion ll~5kJ4 -5 ~), it 18 possible to use its
asymptotic representation Z
Now considerring that
.Z t +i2 11~~"d J~ cGS (?c-M1)=k, R`, (Gas xoGrs )(.F-S' X0' j"
P =k, (zr-- l-n~(z+x,)) I~
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7,
COL
This makes the following tabulation of the function F (-~7) possible.
~Z
I F(j)
~argF
?
AFC )I
ar F
0 1
0.25
1.12
1.6
O.62
0.48
0.2
0.36
1.00
1.9
c.84
0.43
0.4
0.51
0.85
2,2
0.86
0.39
0.7
0.65
0.71
3.0
0.89
0.30
11C
0.72
0.61
4.0
0.93
0.25
1.3
0.78
0.53
5.0
0.94
0.22
In tho p'...nc r,z wh r y be 00=_ frM (35)the ? front of a ewnondnrv
wave ie rectilinear (and will beconical in opace, because of the cylindrical
symmetrical problem and normal to a straight line running from the imaginary
source at an an 1e ,' , as shown in Figure 4 plotted in approximately the same
way as by Ott.L992. The lower edge of the front of the secondary wave coincides
with the edge of the front of a wave propagating in the lower medium with a veloc-
ity ci (ci> co). Its upper edge merges with the front of a reflected wave.
Variations in the amplitude of a secondary wave in prolo ton along the
dividing boundary are determined by the basic function fF (,)~/, ? (Figure 5).
When increases, which ccrreepondp to prolongation from the dividing boundary,
it increases continuously tending toward a constant magnitude (in our article
(loJit was erroneously stated that. when X + , the amplitude of the secondary
wave tended toward zero) when % --lX , (see IV).
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7.
A secondary wave, dust as in the second term of (17), is on the right
side of the first term in (17), obtained by an approximation from geometrical
optics. It is evident, if only because f. -' 0 (ko- oo}~ r
which corresponds to transference to geometric optics, that the amplitude of
a secondary wave in (35) reverts to zero. Nevertheless, as, in gecmetri; optics,
a secondary wave may be graphically represented by imagining that in addition
to ordinary beams reflected from the boundary at angle of elevation X
(beam QOP in Figura 6) there are also beams incident on point P from point 4,
which fall on the boundary at the angle of complete inner reflection xa,
then propagate at the lower medium all the dividing boundary and, fimally,
are re-emitted in the upper medium (beam QrDP). Here, too, X must be smaller
than '(d. It is easy to prove that the calculation of the optic length of such
beams gives a correct phase value for secondary wave and defines its front, It
may also be demonstrated that the beam QCDP satisfies Fermi's principle Laic;
7r
Feivasc?] that the time of its track from 4 to P will be less than all 9th
possible beams the paths of which lie partly in the lower medium (a is L 3J ,
Chapter XII, especially p 528, by S. L. Sobolev ) This time will also be
less than the time of the track along QOP.
A secondary wave admits of graphic interpretation from the wave view also.
Who.-, a wavo omitted by source 4 falls on the dividing boundary at an angle of
compl.eto inner reflection, thens ib developed in the lower ue&tum a wave moping
along the boundary and creating a disturbance in it with it spatial period equal
to. the wave length in the lover medium. This distance conZitionc.the appearance
of a new wave in tho upper medium. But aince the wave length of the latter (A p)
is lose than in order to have periodicity along the boundary
with a period it must be so inclined that the normal to its front forms
the angle X. with its front. The front of r+ secondary wave is located exactly
in this manner.
The secondary wave is of importance not only as a correction for geometric
optics, but also in eeismanetry. It must ob7iously appear in the propagation
of sound Impulses from the sea. I. plays an important role in the propagation
of radio waves in land transmitters and receivers. In the latter case, primary
and reflected waves are generated in the earth and fade very rapidly. A sorordery
wave remains, which passes into the air fros the earth, proptFAtea there and falls
back on the earth.
Schmidt's [7] qualitative rules for secondary waves are in agreement
with our theoretical results.
Ott's ['9_7 expression for this move is represented by the dotted line in
Figure 5, and coincides with thn curve when lY .0. 1.
When. n > 1 (the velocity in the lower medium is lees) the secondary wave
will occur in the region )(< arc cos . The analytical expression for it
corresponds with (35) when F (P ) = / . This is clear since now?(,, is imagimry
:nA_dronaseuently. Y cannot be near Xn_ Hence here ] >> 1 always and, hence,
in accordance with (37), F (lr )= 1. Putting n> l in (35) we obtain a wave
with a front normal to the dividing boundary, propagating along it with a velocity
Cl and fading oxponantielly in the direction of the z - axle.
A partial theoretical analysis is also to be found in the work of Jeffries
L_13.].
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IV. THE REFLEO.TED WAVE AT ANGLES NEAR THY ANGLE OF FULL IN M Ii&ti'L CTZON
To study reflected waves (Pi (17) in the region of angles near Pr , we
shall start from expression (11) and discard the term 1/81k r?eos0( in paren-
theses, as it is small. Integration proceeds along r (Figure 2). The
point o(- ~?an3 the small region~~ near it are critical. Near of the
radical gs be ama1i.'a td hence the expression behind the ix.tegra?
sign rosy be expanded into a series as far as the second term. Introducing a
new variable (9aro(?-)(0 and taking (2 ) into account, we obtain.
0,i sin 0(--g = .2
~iY Sun oA -+-g /- ~.+sinx Vein 2x?s
0
.2 j/,snf
sinx'o
as co ed with unity, it Is
In order to dieregazd the ratio in
necessary that
The integration path In the plane IS must cross from z -i. -im through
0 to - 2: + Pd a' + i ? Making use of the smallness of g and expressing
the obtalned integrals by means of (30) through a Weber ft notion, we shall
similarly obtain
rr R ~ ?x I'sinxo -~2 ~ _ 2 2
where the plus sign refers to the case X< X (Figure 3) and the minus sign to
the case `X > XD (Figure 2).
when ?X>X,, expression (41) is the complete expression for the field, since
there is no secondary wave 9) in this angle region. The linear combination of
a Sieber function of the order of - + in square brackets is joined with the
__~, of +he. m?Anr of / 12. page 155We now have
'Relba 3~; n
?'` qr= a~R + 8? /_e4 .~ !-1)YT (i'P- (42)
R rJinX~ i
The eign of the rm&iar1 sip p meet be taken as different on the different
lsyexs, like the sign of q, inasmuch as when X > )(0 the path r, on the
lower half-plane c-)ssae along the lower layor.
Ix-x/I/x,11 all toms in tho square brackets may be disregarded and the two
Ttwztning torma in the outside brackets will be the two first terms of the
expansion of the usual coefficient of reflection (19) in a series in
4o=- ,g'X Ci/sin2ZsIrh~% n hypothesis (40). In this case, 041
amounts to (17). Hence for the validity of the letter, the c-.ndition
v
must be satisfied or
koR(x-x,)1~0' i.
Cation of D (v -J j ) also has
n
e
ti
t
e
c repro
o
When > : the asymp
coincides with (44)
another form and contains two terms, one of which eJct1jr
and the other gives a wave with a conical front (35).
The reflected wave 19k can now be described for all angles.. When condi-
tion (45) is satisfied, It is given by (17). On the other hand, if this condi-
flon is nnt satisfied. it means than k?R (".)C-Xe )2~ 1. We can write the
latter by virtue of (38), as "1~lF L. aatisfacUuu of
this cothition perliita expansion (39), whence a ession (41) or (42) may
be obtained.
PAIL
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reflected wave the latter the secondary wave Fr ;
for geometric optics, but we shall give more precise values for the integraib in the
theory according to the ?pass" path and the profile margin. The former gives the
In the study of further apprcacimatione we shall not find any new wave types
like the secondary waves given by the second approximation in the formulae
Assuming first that )n/ > 1 and substituting r.2 for qo2 n..1-w ain2 7( ,
(this assertion has bee corroborated by calculations not given hers).
INI4k?R.
ha )2-
i
j
21i.r In
Orif sin) Jn ) '". (48)
I. eleetrodynamice where m . n2 this will. be
which signifies t. t 'the dietancne must be ,;eat. This condition cannot be
fulfilled when /121 Is large, but it is then ossible to use the Weyl-van der
Pol formula ([3~, p 954; L 14J, p 1.05.). From a purely mathematical view-
point, the similar convergence of a series in r , when In/ is large (or
M - P) c A.
iy - in acoustics) depends on one of the parts of the expression
behind the integral vign studied above heIng situated near a point on the
patL of integration, as a result of which the ordinary "pass" method would
be inapplicable. Ott L 15J and Fok [162 generalized the "pass" method,
taking this circumstance igto account and thus were about to obtain the
Weyl-van der Pol formula, but we shall not stop there.
Condition (49) is fulfilled in most cases of radio wave propagation.
Since secondary wavers' (fading as rapidly as waves i the earth) cn be
disregarded when the source and receiver are located in the air,the Hertz
vertical vector component of a reflected wave will be fully given by (17).
Adding a primary wave, we shall obtain the total value of the above vector
In the upper mediumjvhen the transmitter and receiver are raised
ask?R? + e tkaR /~2sinY- (50)
R R 1_~i'-sinX+
where it is assumed that sin2'x can be disregarded as compared with n2 - 1.
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Components of the electromagnetic field are obtained from (50) by ordinary
differentiation. This expression is not new in electromagnetics and i? derived
in meet cases from Wise's series, but as far as we know the expression for the
corrected term of N given here has a wider ei ificance.than anywhere else.
In addition to tie cases in electrodynsnice and acoustics mentioned, the
"pass" method has a limited application when n is so close to unity that condi-
in an approximation for geometric optics (acoustics) the field in the upper
f which may also be represented as origipating from an imaginary
transmitter Qi (Figure 1). Trig representation is oft.,n used in eleetrodynam-
boundary Ro = H, and the condition of the smallness of the corrective term in
(17), as compared with the sum of the primary and reflected waves may be
written
t,Rl/+f NO)/>INj- (52)
Hera we do not take account cf the cor-?ection introduced by the seconde.7
wave, as it does not change the order of magnitude on the ri ht-hand side of
the latter condition. Considering that, in accordance with (19),
yr,sir
/ (531
and also that R sin z n, condition (52) may be rewritten in the form
When 17,l > 1, the order of magnitude N is given by (47) and the latter
I.fI 3%"A:~4;1, then
Ind{- 14 - pp
WgNJINL
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A
that is,
CONFIDENTIAL,
/jz } /NJ in electrodynamics
7.'hue, for applicability to geometric optics it; is necessary that the eleva-
tion of the receiver (transmitter) on the dividing boundary for the particular
case in electrodynamics where,/ / 1, Vvedenskiy's ([T4J p 113) analysis
of applicability to geometric optics is correct in a completely different form
than ours..) be sufficiently great in comparison with the wave length. With
strongly reflecting,; boundaries, when jFjsin Y y ', it is lossible to use a
weaker condition, also derived from (55 ;
which gives:
in acou3t.c8
The latter ccndition6 permit making a limiting transition to the case of
absolutely reflecting boundaries, corresponding in acoustics to IP --ef -+ m
in eleotro-3,pnam'ce to 1-ml --> oo . In both cages, the right-hand aIdea in (59) tend
toward zero, whence it follows that geometric optics is correct in this caps
for any z.
Two genernliza.tione of the results obtained axe-
a. When 14141 in electrodynamics, the conditions will be written like (55),
(56) and (58); in acoustics n will appear instead of = . This is easily ob-
tained.from (54) and (20), if it be considered that, keeping the correct order
of magnitude, it is now possible to discard n2 apd) ' 3s sin2 X as coapared
with unity, and in consequence, for embtple, n2- 0082 'X a n2 - 1 t sin
may be replaced by - 1 and so forth.
b. In raising the transmitter and receiver, it is sufficient for correct-
ness in geometrical optics to require fulfSllment of the above-mentioned condi-
tivru, except that it In no canary to rep?_ace a by (the total eleva-
tion of the transmitter and receiver on the dividing boundary). Here sink-4
Graphs can be mach from the study of the reflection of spherical waves given
in the preceding paragraphs r/~ 18_7. It appears, therefore, that of all plane-
wave rays given by integral (4), on the boundary, only those rays of which the
elevation angles are included in the small region of the order of near'
angle )( play an important part. It was likewise found that no all dividing
boundaries are of importance in the reflection of a wave, but only certain
"effective zones," elliptical in form.
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CONCLUSION
We encounter the problem of spherical wave reflection in the study of
the field around a point transwitter of 0,)und or of eleotronagnetiu waves,
placed on a plane boundary of separation dividing two media. This article
given a detailed study of a reflected w?ve in a wave zone by expanding solu-
tiouP into. series in . Vkr , using the "pass" method L Scemmerfeld'a method
.of complex integration). Thereby practical formulas are. obtained for the
calculation of aHertz vector in electromagnetics and of a sound potential in
acoustics for any-height-of elevation off the transmitter and receiver over
the dividing boundary. Moreover, a full dear-iptioi.isgiven of a new type
of vavs (a secondary wave) which,plays an, important part in mart'-cases. As
compared with the p'ibliehed: works of Ott and Srtger, a full descri-ratios is
given here of afield which is useful at all points of space in a wave zone.
.In addition, our article gives a complete analysis of a solution by which
criteria are found for applicationsto "gecsletric optics. It also explains
many other important peculiarities of the problem.
I consiiew it my duty to expreusmy deep gratitude to Academician V. A.
?Fok for much valuable advice.
- 16
COAL
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BIBLJCOGRAPHY
1. gomaerfell, A., Ann de Fhye, 28, 665, 109; 81 1135, 1926
2. Weyl, H., Ann de Yhys, 60, 451, 1919
3. Frank,.F. and Mises, L. Differential and Integral Equations of
.bathematical Pbreics, ONTI, Ch XXIII, edited by V.A. Fok, 1937
4. Ler*ntovich, M.A., Izd, AN TSSR, Phys Ser VIII, 16, 1944
5. Schuster, K., Akust, ZS, 335, 1939
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13. Brekvo,?skikh, L., :1FN - 32, 464. 1947
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reflected wave
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Figure 3 Figure 4
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