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Supplementary remarks on reflection and deflection of sound in water online

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*«W YORK UsIIVERSITY

•ermjTE of mati-cmatical scsences

LIBRARY
4 WtJbioffoa WHre, New Yo»k J, N. Y




A.VG'-imT No. g

*"~~" NEW YORK UNfVERSITY

INSTITUTE OF MATHEMATICAL SCIENCES
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^, J^iict^io. ^^f / -M ^ "^^^'"S'- "-'' New VoH. Z, N. y

^ Supplenentary Remarks on
T?©flectlon and Deflect Ion
of Sound In ' 'ater

by R, Coupant and K. Priedricha



In a previous report reflection of plane sound waves

In water was studied under the following assumptions:

The velocity of sound Is a function k(2) of one coordinate

2 only, and k = k Is constant for large positive and large

negative z, while In an intermediate layer of thickness

fiX(X s wave length) k may differ noticeably from k with an

o

extreme value k^fl + (T ). The coefficient of reflection
was discussed as a function b(9) of the angle of Incidence
between the norral to the wave and the planes z = const.
Various plausible velocity distributions kfz) were considered.

In this supple.ment three different points are
discussed:

A) A simple velocity distribution (k = k for z < 0,

o '

k = k^ for z > ¥Xf k = k^(l + C" ) In the layer between).
o o ■ '

already mentioned in the previous report, is here studied
In detail.

B) A method for treating reflection on moving layers is
developed.

C) Deflection of sound along curved interfaces is
estimated.

A) Haylelgh*a Velocity Distribution

Graphs for the coefficient of deflection |b(9)| are given
for the values ^S" = + .001, + .005, + ,01 and for layers of
the thickness V = 2, 4, 8, 16, 32 wave lengths. A comparison
with the results of the previous reT5ort shows a behavior



fTKs^vv., iiO. -van

ZXJ^IX JADITAMJUTAAA lO HTLHTreMi

YSAaatJ

Y .H ,t i*ioY waH ,i3r.!*l riolgpMjf.W ♦-



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qualltatlvely the serre as for gradually chenglnp: velocltlea
If sT!7ftll angles 3 ere considered, T^oughly one night
Idertlfy the nixinber N of the "rain case" of the tsrevious
report "



-*u>



is the same as for fluid at rest when it is referred to a
coordinate system irovinp^ with the stream. Hence when P
as a function of x,y,z,t is put in the form

P = P(x - trt,y,2,t) for z > ,



•-J «.



* lo



■^ *-



^



■Oi 4-



*
£



-4-
the wave equation la, as befoi*e.

The transition conditions are the same as before: P and ^

825

are continuous at the sixrface z = 0. The nressure P for

z < Is ap;ain represented by an Incoming and a reflected wave,

P = ^i-(-^^'^^'^t) ^ ^ ^l(ox.Y2-6t) ^ 2 < 0,

uhlle the transinltted wave la

1 (a, C x-Ut ] +Y, z«0, t )
P = ce ^ ■^,z>0.

The velocity of sound, k. Is assumed to be the saire In both
fluids.

The wave equation yields the relations

a^ + Y^ = k"^6^ and a^ > y^ = k"^0^ .

The transition conditions laiDly that the exponents In the
exponential functions coincide for z = 0, l«e«

furthemore they yield

1 + b s: c, y(1 - b) = Yn®*

These relations may be rewritten in terms of the angles of
Incidence /O, S). , frlvon by



V



rrrfrt r^'S'frf



.Oj-Y









:W



.•T'



COS



-5-



9 = k a Q , cos 3^ s k a Sj-"
slTi = k Y S"^, aln 0-^ = k \^Q^^ .



Then we have

sec ^^ = sec J - or ,

with ^ = k'^J, Purthernore

b = sin( ^ - S, )/sln( 3 + *^ O ,

c = 2 aln i) cos ^ ./8ln( ^ + '^^),

These relations may be compared with those for the case
that the fluid is at rest for z > but has the different
velocity

k,^ = k(l - 2 T)"^/^ ,

where sec '^i = s®c J (1 • 2 (r) ' , while the expressions
for b and c are the sare (coTpare cases 1 and 2),

It is to be noted thPt the case U > corresponds to
k, > k; I.e. the case whore the x-component of the velocity
of the Incoming wave falls in the direction of the moving stream
corresponds to the case where the velocity In the secot^ layer
is larger than that with which tl^ waves arrive.

The rosults for both cases nearly coincide In the vicinity
of the angle of total reflection wher« sec D^ = 1, or at
glancing angles, X) -^ 1, sec /J '^l.

The case of a layer of finite thickness, < z < k,
moving with a constant velocity while the fluid at both sides



■^Ts n Hi s



fjy^ti. mf r.^t



v. D



.t.



)V



o:* ''■ o& Bx :^1

yilnloiv ©1!:+ ni

. " « -^ ■• ->* "rto ' lo



-6-



Is et rest, can be treated elpllaply. The resulting forrnilaa
ij!>e alrllar to thoae obtained for oaae S In the rsrevloua report.

C. Influence of Curved Interface a




The simplest case of a velocity k which does not
depend on one coordinate, z alone la that of surfaces of
constant velocity very nearly orthogonal to the z-axls. In
thlsi case perturbation theory is applicable. The calculations
are not reported here, since they have not, so far, yielded
quantitative results. However, a surnnar^-^ of these calculations
wlliil be presented later after completion of the work.

Curved interfaces deviating considerably froir, planes
soem much nore significant for the actual phenor^ena. According
to a suggostion by Dr. H# '3. King, one might expect large
deflections by repeated reflections of sound along such surfaces.

The follo^'.lng arguisont, based on the previous results,
gives a rough estimate for the sound conducting effect of such
Interfaces or layers. The effect seems large enough to
account for bending of sound beair^s throu^ any angle, in
particular when the sound has to travel through an area In
which many such interfaces are situated at rendon in varying
positions. Our asaunptlon la: The portion of the sound beam



• Ci"»



« ^"£0&^



■:. la :



-7-



under consideration ray be regarded as a piano wave, and
the curvature of the Interfaces should he sroall enough to
permit the anpllcatlon of "geometrical optics". In other
words, we consider the beam as a ray, and we assume that
reflection end transmission of such reys can be calculated
as for plane waves, so that the coefficient of reflection
1b given by our previous i»e suits.

"^Je asiune as a typical case a surface of discontinuity
as in the diagram • The path of the ray Is indicated,
where the angles which the polygon makes with the tangents
are -S^^* -^q'*'* • '^K* '^^^'^ ^^ total deflections Is



a = 2(^^ +S)g + ... +B^).



If the aiiiplitude of the incoming sound is 1, then that of
the last reflected beaw will be the product

If b(9) la the coefficient of reflection obtained for plane
interfaces with a corresponding distribution of velocity
k(z). Now all our previous results show that b(9) has the

form

b(9) = 1 - e £) ,

irhere e = e(9) is sinall for small angles ^ . Now

p = I (1 - B^O^)il - t^9)^)...{l - e^^)|

can be estimated by taking the logarithm, which will be
approximately



.h tc



■'Ota 6rf:t Of



•*s'^ *l"^



II



-8-
log p ^ ' (^2pl "*" ^2% -^ ••• + ^nV



»



ot If all e. are less than a snail number e,

log P > - -^ •
Thus approxliretely

p > e-^V2 .

Prc3tn this It follows that p will approach unity If the
angles z)^ become very small, while H (for the given total
an^e c of deflection) tends to Infinity. In other v/orcia,
the neai'or the first angle of incidence cones to z°-ro and
the more frequent the single reflections become, the more
total the deflection of the sound beam by bending along
the Interface will be.

As an exarrnle, assTime S a half-circle. All an.c^les ^
will be equal and -^ - -^ • Y'e assume that in a strip

of the width of 4 wave lengths around our half -circle the
velocity is k(l + C) with (T = .005, and 3= 3°, R = 30.
Then fron our graphs we firwi z < .001 and consequently



will be approximately larger than — . In other words, almost

e

half of the amplitude of the incoming sound is deflected by
180 degrees.

It should be noted that the effect of deflection will
be slightly enhanced by deflection of the sound transmitted
into the intermediate layer on its other boundary.



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l«WYORK UNfVERSITr

HnrruTE of mathematics scjencb

LIBRARY
4Wawtf Pbce. N«ir Yo«k 9, H Y





1

Online LibraryRichard CourantSupplementary remarks on reflection and deflection of sound in water → online text (page 1 of 1)