Richard Courant. # Supplementary remarks on reflection and deflection of sound in water online

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*Â«W YORK UsIIVERSITY

â€¢ermjTE of mati-cmatical scsences

LIBRARY

4 WtJbioffoa WHre, New YoÂ»k J, N. Y

A.VG'-imT No. g

*"~~" NEW YORK UNfVERSITY

INSTITUTE OF MATHEMATICAL SCIENCES

LIBRARY

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

^ Supplenentary Remarks on

T?Â©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 > Â¥Xf k = k^(l + C" ) In the layer between).

o o â– '

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 â™¦-

-2-

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 > ,

â€¢-J Â«.

* lo

â– ^ *-

^

â– Oi 4-

*

Â£

-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, zÂ«0, t )

P = ce ^ â– ^,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, lÂ«eÂ«

furthemore they yield

1 + b s: c, y(1 - b) = YnÂ®*

These relations may be rewritten in terms of the angles of

Incidence /O, S). , frlvon by

V

rrrfrt r^'S'frf

.Oj-Y

:W

.â€¢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 = sÂ®c J (1 â€¢ 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 wherÂ« 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

â– ^Ts n Hi s

fjy^ti. mf r.^t

v. D

.t.

)V

o:* ''â– o& Bx :^1

yilnloiv Â©1!:+ ni

. " Â« -^ â– â€¢ ->* "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

â€¢ Ci"Â»

Â« ^"Â£0&^

â– :. 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 iÂ»e suits.

"^Je asiune as a typical case a surface of discontinuity

as in the diagram â€¢ The path of the ray Is indicated,

where the angles which the polygon makes with the tangents

are -S^^* -^q'*'* â€¢ '^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 Â£) ,

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

â– 'Ota 6rf:t Of

â€¢*s'^ *l"^

II

-8-

log p ^ ' (^2pl "*" ^2% -^ â€¢â€¢â€¢ + ^nV

Â»

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

log P > - -^ â€¢

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 zÂ°-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 -^ - -^ â€¢ 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= 3Â°, R = 30.

Then fron our graphs we firwi z < .001 and consequently

will be approximately larger than â€” . 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.

/ -^

L-'

ee

e

^ e.

â– \M^'

â– . .i- r7j '. I

'iJC .'. -â€¢.'

, A?

w

ra

HH

:i

ft

d.

a

^

13

â€¢-t

U3

o

W

sr

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"1

O

7s

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a

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re

â– -â€¢1

>

a.

>

o

o

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PI

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cn

O

â–º1

O

7s

-n

9

(0

n

z

o.

m

S

s

^

o

o

a>

CO

4^

en

â€¢-1

TJ

13

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33

M

lÂ«WYORK UNfVERSITr

HnrruTE of mathematics scjencb

LIBRARY

4Wawtf Pbce. NÂ«ir YoÂ«k 9, H Y

â€¢ermjTE of mati-cmatical scsences

LIBRARY

4 WtJbioffoa WHre, New YoÂ»k J, N. Y

A.VG'-imT No. g

*"~~" NEW YORK UNfVERSITY

INSTITUTE OF MATHEMATICAL SCIENCES

LIBRARY

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

^ Supplenentary Remarks on

T?Â©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 > Â¥Xf k = k^(l + C" ) In the layer between).

o o â– '

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 â™¦-

-2-

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 > ,

â€¢-J Â«.

* lo

â– ^ *-

^

â– Oi 4-

*

Â£

-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, zÂ«0, t )

P = ce ^ â– ^,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, lÂ«eÂ«

furthemore they yield

1 + b s: c, y(1 - b) = YnÂ®*

These relations may be rewritten in terms of the angles of

Incidence /O, S). , frlvon by

V

rrrfrt r^'S'frf

.Oj-Y

:W

.â€¢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 = sÂ®c J (1 â€¢ 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 wherÂ« 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

â– ^Ts n Hi s

fjy^ti. mf r.^t

v. D

.t.

)V

o:* ''â– o& Bx :^1

yilnloiv Â©1!:+ ni

. " Â« -^ â– â€¢ ->* "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

â€¢ Ci"Â»

Â« ^"Â£0&^

â– :. 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 iÂ»e suits.

"^Je asiune as a typical case a surface of discontinuity

as in the diagram â€¢ The path of the ray Is indicated,

where the angles which the polygon makes with the tangents

are -S^^* -^q'*'* â€¢ '^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 Â£) ,

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

â– 'Ota 6rf:t Of

â€¢*s'^ *l"^

II

-8-

log p ^ ' (^2pl "*" ^2% -^ â€¢â€¢â€¢ + ^nV

Â»

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

log P > - -^ â€¢

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 zÂ°-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 -^ - -^ â€¢ 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= 3Â°, R = 30.

Then fron our graphs we firwi z < .001 and consequently

will be approximately larger than â€” . 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.

/ -^

L-'

ee

e

^ e.

â– \M^'

â– . .i- r7j '. I

'iJC .'. -â€¢.'

, A?

w

ra

HH

:i

ft

d.

a

^

13

â€¢-t

U3

o

W

sr

3

"1

O

7s

JO

a

>

re

â– -â€¢1

>

a.

>

o

o

X

PI

1

cn

O

â–º1

O

7s

-n

9

(0

n

z

o.

m

S

s

^

o

o

a>

CO

4^

en

â€¢-1

TJ

13

3.

f

w

"1

p

r

3

o

f

<

o

3

â€¢^

o

rt-

fp

m

n

e

P

?>

3

O

8'

P

3]

33

M

lÂ«WYORK UNfVERSITr

HnrruTE of mathematics scjencb

LIBRARY

4Wawtf Pbce. NÂ«ir YoÂ«k 9, H Y

1

Online Library → Richard Courant → Supplementary remarks on reflection and deflection of sound in water → online text (page 1 of 1)