SCIENTIFIC - PHYSICS

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP80-00809A000600230675-8
• Release Decision: RIPPUB
• Original Classification: C
• Document Page Count: 10
• Document Creation Date: December 22, 2016
• Document Release Date: July 15, 2011
• Sequence Number: 675
• Publication Date: June 24, 1949
• Content Type: REPORT

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DATE OF
i.IFC^.MATION 1948
DATE DIST. 'ice Jun 1949
NO. OF PAGES to
SUPPLEMENT TO
REPORT NO.
THIS IE UNEVALUATED INFORMATION
Zhur1 1 Tokhxaloheekoy Ris1kl,,9ol XY-Lll[, No 4, 1948.
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COUNTRY
GLAVICATION
CENTRAL INTELL
FOREIGN DOCI'MENTS OR RADIO BROADCASTS
SUBJECT Saientlf:.o - Physics
HOW
PUBLISHED Monthly 4r,riodLLal
WHERE
DATE
PUBLISHED Apr 1948
LANGUAGE Reiseian
L. Brekhovskikh
Phys Tast Aced 801 US8R
i7 November 1947
(A11 figures are appended .7
? This is a study of the field of a point radiator emitting sonic or
radio waves over a flat boundary of separation in the came where the prop-
ertiee. of the media on both sides of the boundary of separation are app,?ox-
imately equal to each other. The study indicates the limitation of appl1ca-
bility of present theories. Rules are found in the fulfills^at of which tral -
ettizmi layers can be replL.ed by bc-andaries of separation.
11 approaches unity at the dividing boundary --
terletios t imtim
L, 6, lj7) Clarification of peoul;ar ehara0 ,
also
' .-A *h. d.~v.loommnt of formulua, iaa
Z7[. eoinL10ne, nv.c.ci, -.- o???-- - ---- - -- -
large values for the amplit.des of reflected waves as the index (4f refraction
the i'irut Zara of r27,
INTRODUCTION
Be meet th7 problem of the reflection of spherical waves in studying
alactrr .tic or sonic radiation on the presence of a boundary surtAce
dividing two media. Ku)i x.)rk has been devoted to this problem, O4yOOi&AJj
from two staadpointu: (a) the study of the field over strongly
dividing boundaries, oor esponding in .loot tins to the propagation
of radio waves over high conductive groundV; and (b) the study
of a field F?b a t4act'on of the various pr,.~-p.rties of the media 5, 6: 137.
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which the index N may approach unity a,_ _ oeely as desirable, are of theoreti-
cal and practical interest. Particular!- in the study of the propagation of
radio waves in the atmosphere, it is often necessary to deal with their reflec-
tion from layers wbioh can be regarded as surface boundaries dividing media
with dielectric constants approximately equal. The index of refraction for such
viding boundaries can differ from unity only by small fractions of a percent
Li7]. The reflection of sound from "skip layers" in the sea is such a case.
The latter, as we knot', represents comparatively fine layers, extending in a
horizontal direction, with large gradients of temperatire and salinity. The
index of refraction, equal to the ratio: of the velocities of propagation of
sound in water above and blow the skip layer, also differs from unity only
by small fraction of one percent.
Hereinafter boundaries dividing media with properties closely related
to each other will be called this.dividing boundaries.
The formulas obtained by ldyazin [12] for the Hertzian vector are used
for this type of boundary, but refer only to the case where the emitter and
receiver are at the dividing bc.mdary.
The particular situation existing when k approaches unity was noted
first by ua;'13]. Also noted wee the fact that the study of reflection by
e pansion into a series in (reached by a method used in the work
of aft s>rfeld jij; it Wei 5] and Ott [6]) is possible for all angles
only when the following
ndition is satisfied ([13 ], equation 28)1
ddo
~
on
VR
where R is the distance from the emitter to the receiver. If this condition
be not fulfilled, the above-mentioned "pereval" method [literally "pans"
method used by Romaerfeld at alii] gives the correct values of the field only
for sufficiently large angles '(Figure 1), satisfying the c9ndition -.
ke Rx~~, ~ .
The inapplicability of thista6thdd when 7s is large and the angles X are
small naa be readily seen from graphs which are of no interest here. An
important role in reflecting spherical waves is not played by all o: the points
of the .Ividimg boundary, but only by a certain elliptical zone flh ] the area
of which tends toward zero in,geoetric optics, when it is possible to speak
of the reflection of a been from a definite point on the dividing boundary.
Consequently the refloated wave at an arbitrary point of space 4epends on wave&
passing only along thobe directions which correspond to straight lines connect-
ing this point with all parts of the effective zone. These will be directions
diverging from the direction of tho beam reflected according tb the !are of
geometrical optics at small angles of the order of ~ or loss. Appli-
ration of the "pass" method is possible- if it can be assumed that, within
this area of angles, the coefficient of reflection and its derivative with
respect to the angle do not change much.
However, from the well-ksawm expression for the coefficient of reflec-
tiou of plane waves from a boundary dividing two media:
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L7(A~ = ^?v irs.}'- J`3z1_.c osZX
Al Ssn X-- dy,a. -`-
_:croyi
we find that, when n approaches unity and X is small, the first and
second derivatives of B ) may be as large as desired. Here m - n2
in electrodynamics and A ?l (ratio of the densities of the media) in
acoustics.
Application of the "pass" method gives the usual formulas of geometrical
optics (acoustics) in the first approximation and their corrections in
succeeding expressions. When 7i approaches unity, condition ,2) guarantees
the smallness' of these orrections. Hance it is the condition that ordinary
geometric op*bs describe reflected wave.
It may be dsionstrated r9 7 that for any eYnvnan 4 ------- --Al.
B. Radiation of a Point-litter Located Above a Ph.D. Dividing BounderY
llgere 1 shows at point (r., z) the radiation reflected from a point-
emitter at a height co (which may be zero) from the plane dividing boundary.
In eleotrce.06A tics, a vertical dipole is selected as an emitterg in acoustics,
a pulsating sphere of infinitely small radius. The radiation field will
be characterised by the scalar function 5Q, which -orreeponds to the vertical
ooaponent,of the Bortzian sector or to sound potential.
where the first term represents a primary wave, the second a reflected wave.
And
a ko R,,
IP=
ku and kl are vave numbers Li the upper and lover media, respectively; mhos
the same value as in (3) and ,f = n+r.o.
Our problem is the study of expression ?) when n : k3 approaching unity.
Let us expand the expression behind the Integral sign (5) into a aeries in
(79% b. To this end we introduce the no~ation
Here bo.- ? b a
xm k,1-koa_ ,E Y j -l (6)
dls 45.
Awn
'Mb -br
(7)
Subseituting this in (5) and takix (6) into account; we obtain for a
reflected wave (the radius of convergence of.s riec (7) equals unity. Hence
representation of 996 in form (8) is poecible only if in (5) a path of inte-
gration is select--d much that anywhere in it 1'X 141 that is Ib" I1>1 ko: k =I
We can, for instance, soleot a path which rung first i,..,: !' = 0 along the
real axis; than by-passes the point s = ko along a semicircle, lying in the
fourth quadrant, with a radius greater than and comes out
spin at the real axio. 31n4, in the four7,h quadrant, the expression behind
the integral a'.an in (5) has no sing-alar point, iraaeition io such a path of
integration is possible. It is ar9umed, moreover, that in integrals (y),
obtained after expansion into a series, the integration path is the real axis,
for which mason tiie values of the integral, of course, do not change.):
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c
sox
s~ to i J. tyr)~; d (9)
For the coefficients of expansion Is in ('j) we shall have
(10)
.whence the consequent., coefficients can be found by means of the recurrent
foriala
d )
(~'+ ' / -F- (js-l ~ ?rr (g5 -l ~ arcs I 11
. study of integral I for any s is give in the appendix. were we
are restricting ourselves to the case where (n - 1) is so small that its
second and subsequent powers may be disregarded in (8). Nov, taking into
account the well-known foraula ((3J, page 94l):
As desanetrated to the appendix, the in-gal Il can be iepresented in
the fora
R, =/r;+t z,
,Q-4 Ca r; D=-1v2 re
The integral tore in (14) must still be studied. Taking the difference Of
the two integrals, we have obviously
eE R, tdt?
(16)
e C C
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To calculate the first Interval we substitute variables in accord-
ance with the formula
Taking the second integral and substituting 'Aho limits gives:
L ( tko ,ei.kar}
ko
Therefore we bbt49.n
low for all integration' paths az aE? p kor and as
term under the square root sign in (18) can be discarded.
/fit-d1=
k,, SRI-.~-}~~r$. w~e~e~---zorda
ikekl
R drm~ ~(~.).t.d5(u)7,? (19)
and CL'n) and g(a) are Iresnel integrals. Substituting this result in (14)
and taking (19) into account, we obtain an expression for Il which, by sub-
stitution in the expression for a reflected wave (1~), gives for the latter:
IL 1
7M i"10 I-rha C-+. is (21)
In electrodynamics, where m = n2, the first tern in the parentheses is
kor times lees an the second and may therefore be omitted. In acoustics
we'are only interested in the came where m - 1 is equal to or lose than n2-1,
so that this first term may again be omitted. (When m-i is not a small quantity,
there will be no singularity when n approaches unity.) A reflected wave
is now written
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r~r
C r 1 50X1-HUM
In electrodynamics the cc:fficien#. can be retlaced
by j (since than 1' equals 1).
Btpression (22) represents the usual spherical wave reflected with a
coefficient of reflection B. It may be shown that when td~'~ 1 this B
coincides with the coefficient of reflection of plane waves (3), if the
lattrr aijo is expanded into a series of powers of (n2 - l} and equals approx
ixttely the first term in the expansion. This is to be expected in view of
the condition iz3 ) 1, which may also be written
(24)
Introdnoing X-k'W instead of tq and using the asymptotic expansion of
'lresnel f8_7 integrals in powers of + , we shall have
ss.U quantities, weehall obtain for " y, - -
B a ?n~/-~saJ - , (25)
/ frml l) l si na ,2
the same expression as found by expanding (3) in powere of (n2- 1), to as
far aa.the first term, and discarding the term ~+e , for reasons indicated
above.
When u4 1 in (23), all terms in the square brackets may be diereaarded,
whiah will again leave the coefficient of $ in a simple form.
It is interesting to study the ratio: ae %runs through all possible
values. e
Figure 2 gives the graph of thlw ratio aq a fiunction of24 , with logarltb-
51o .Dales on the axes. A*'Lcincreases ,)L)increases continuously. At first
this growth fo:l.lc'w the. Bay for vg1, which eorrespondu to a straight line on
the gram soals. When is in large enda toward unity, which is the condition
for the applicability of geometrical optics; that is, the ooeffit, _ente
reflection of a Ocpherioal and a piano wave coincide.
is result refers to the case of the fairly fine transitional layer
treated in 4 lalove. For oases of thick transitional layers only the reflec-
tion of plans raver 'ins been oalculated;[7J, but '.t is possible that condition
(24) my also be used in determining these cases.
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Co
Condition (24) cannot be n-3ed for ?;?aiaple, for the case of radio waves
near a thin dividing boundary 400 meters above ground and 50 kilometers
distant, when k? 'equals 10 for A:- 2 meters and 2 for k= 10 meters.
In the latter case., the field of reflection of a wave must ha calculated
by formulae (22) and (23).
In hydroacoustioa, when the depth of the skip layer is 30 meters and
distance R equals 2 kilometers, a reflected wave cannot be calculated
according to geometrical acoustics when )~> 1 miter.
For practical conditions it is interesting to study the propagation of
sound, and radio waves in a layer limited on one side by a thin dividing
boundary. We have previously shown [9J that in the most general case a
field in a layer may be developed for an infinite number of waves reaching
t# I e receiver, after a number of reflections from the boundaries of the layer
this casee the amplitade of the wave oxperienoing k reflections from a thin
dividing boundary will have the coefficient (n2 - 1)k; whence it follows thaty
in our approximation, it will be necessary to calculate the following six
waves which acre not reflected or are reflected only once from this boundary:
(1) primary; (2)." reflected from the usual dividing boundary; (3) reflected
from a thin reflecting boundary; (4) reflected first from the usual and next
from the thin boundary; (5) the same in reverse order.; and (6 reflected
twice from the''usualcnd once from the thin dividing boundary.
The second wave can be calculated by earlier formulae [13] The fourth,
fifth and sixth can be reduced to the third, if the usual dividing boundary is
absolutely reflecting, like the surface of water in hydroacoustice. Here the
amplitude can be calculated by means of (22) and (23); (20) will be true for
if story path taken by a corresponding beam is projected on the 2-axis
and if It - M -? 4 Moreover, if the coefficient of reflection
$e for an absolutely reflecting boundary is Bo a -1, as in the above bydro-
acoustio case, the amplitudes of the fourth and fifth waves must have the
minus sign. When the second boundary is not absolutely reflecting, the
problrm in more oomplic ted, but may be solved by expansion into a series of
powers of (n2 - 1) and application of the "pass" method.
P. 8ubetituti.2I a dividing Boundary for a Transitional Layer
Ir. practical oases, where there is generally a transitional layer
instead of a thin boundary, the question arises how many such cases can be
reduced to that of a single dividing boundary. It is known that this can
be done if the thickness of the layer is small in comparison with the wave
length. In this commotion also, thin boundaries have a singularity, since
the corresponding condition proves to be "weaker." To deduce this, one
makes use of certain graphs.
Let us examine the reflection of a plane wave incident at an angle X
to a +sansition layer, with thickness 1, which is parallel to the plane ; = 0.
When 2 equals or exceeds the wave length X and the difference between the
velocities of propagation at the upper and lower boundaries is small, the
propagation of a plane wave may be calculated according to geometric optics.
so that the function o)' the wave phase in terms of the coordinate Z will be
i ,on by exp & f Ic 5dz7? The extent of the layer in the direatien can be
g
disregarded, if the pa of the phase in the layer is snail, that is, if
C01L
50X1-HUM
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We shall intensify this condition by substituting for the magnitude ka
its maximum value in the layer. It can now be written
zbwax1