Structural Acoustics:A General Form of Reciprocity

Principles in Acoustics

K. Case

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Structural Acoustics: A General Form of Reciprocity Principlesin Acoustics

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K. Case

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13. ABSTRACT (Wasmum 200 wore)

A generalized Reciprocity Principle for Acoustics is obtained. By specialization,various principles which appear in the literature are obtained.

14. SUIMECT TERnS IS. NUMBER Of PAGES

Rayleigh, mass conservation equation, green's thereom t6. PIKE coot

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Ž6ro t AN % M

Abstract

A generalized Reciprocity Principle for Acoustics is obtained. By spe-

cialization, various principles which appear in the literature are obtained.

1l.,

1 INTRODUCTION

According to reference [1], Russian measurements on the acoustic re-

sponse of submarines to internal and external excitation make extensive use

of the "Reciprocity Principle." While this can hardly be anything but an

application of Green's theorem, the way the theorem is stated seems to be

slightly different than that in the usu l literature - and no derivation is

given. Accordingly, it may b, useful to see what forms of a "Reciprocity

Principle" one might have.

The standard literature is a little vague. Rayleigh's [2] form relates the

pressure at a point B due to a source at A to the pressure at A with the same

source at B. The derivation assumes a bounded region with rigid boundaries.

It is indicated that the results hold for an infinite region with rigid finite

boundaries and a radiation condition at infinity.

Landau and Lifshitz [3]' discuss only an infinite region with no finite

boundaries. The statement is the same as Rayleigh but there is a generaliza-

tion to inhomogeneous media. Addition [3]b there is a problem given which

relates velocities due to dipole sources. Morse and Ingard [41 give the same

result as Rayleigh but for both rigid and compliant boundary conditins.

The result implied in [1] seems to relate the pressure at A due to a localized

force at B to the velocity at B due to a mass source at A.

Below we give a very general relation which requires only an impedance

type boundary condition on finite boundaries. By specializing, we immedi-

ately obtain the three kinds of reciprocity principles discussed above. Clearly

other specializations will yield additional principles. We leave it to the reader

to determine these - and their usefulness.

2

2 BASIC EQUATIONS

These are:

a) The Euler Equation

P{l+a(V v)V} =-VP

b) The Mass Conservation Equation

ap5t+V.pfy=O

c) An Equation of State

P = P(p).

For acoustics we linearize putting

P - Po + P'i, = 'o + f"'.

Here, as compared to Landau and Lifshitz [3), we choose the homogeneous

case p. = constant and i0 = 0. The linearized equations are (after dropping

the primes)

Po 7j = -VPopa-+poV-v = 0.

N--" = '9" The resulting equations are

Po-: = -VP

3

1 oP2- t6ý-- + P0 v . 0.

Assuming simple harmonic motion, i.e., all quantities - e-iw' yields.

-iwpot3 = -VP

-p+ po V 6 = 0.C2 .

Let us introduce forcing terms into both equations. Then

- iwpo0 = -VP + P(i - f1 ) (2-1)

-iWP + poV . I = M(f - ý2 ). (2-2)

Eliminating 5 we obtain

(V 2 + k 2)p = V. P + iwM, (2-3)

where k2 = W3/C2.

We want to put this in a form to apply Green's Theorem. We con-

sider pressures P1 (corresponding to an FI, M1) and P2 (corresponding to a

F2, M2). Thus, we have

(V'2 + k2)P 1(F') = V' F1(i' - •) + iwM1 (i' - i) (2-4)

(V' + k 2)P2(,') = V' .• 2(f' - D) + iwM2 .(#' - F2 ) (2-5)

Multiply Equation (2-4) by P2(i') and Equation (2-5) by P1 (iý'). Sub-

tract and integrate over the fluid. The result is:

Jd3i (P 2( I')V* P&') - P-I-)V'2P2fP)}

4

= d3r' {P,(F')V'.(' - I) + iWP 2 (f')MI(i:' - r)} (2-6)

- fd3r' {Pi(P')V'. -( ) + iwP&(2')M 2(' - r 2)}

Using Green's Theorem and assuming impedance boundary conditions

(i.e. P and RP-- are proportional on the boundary) we see that the left side of

Equation (2-7) is zero.

Thus

f d r'{P2(P')V'. • (?' - f') + iwP2(ý')Mi(F' - ri)}

= ,d3r' {IP(F')V'. •(F' - •)j + iwP,(P')M2(P'- r2)}. (2-7)

This is our general reciprocity principle. Specializing the Ps and Ms

will give the specific principles mentioned in the introduction.

5

3 SPECIAL CASES

Here we will only consider the situation when the Fs and Ms are con-

stants time 6 functions. To keep things straight, it is convenient to introduce

a special notation. Thus

P1 which corresponds to f'(• - i ) equal to F16(f - ') (PA now

a constant vector) and M1 (- - i1) equal to Mb(9 - 72) will be

denoted as

a) Let P, = 2 =O.

M, (f' f,) --- M6(f,'- fa

M2(f -r2) M~' 2

(M a constant).

Then Equation (2-7) reads

P(P" = 0M, Ul,; 2) = P(F = 0, M, 2; fi). (3-1)

Since both sides are proportional to M, we can take this equal to 1 and

Equation (3-1) is the statement that

P(FI; f 2) = P(f2; ý1). (3-2)

7

This is the standard Rayleigh form of the reciprocity principle. In

words: the pressure at ij due to a unit source at i 2 is the same as the

pressure at f 2 due to a unit source at f.

b) Let M1 = M2 = 0.

FA(fi'-) = 2b(fr ý

Equation (2-7) becomes

P1 - V'iP (P2, ';i~ f 2I V' P (P1F'i;i (3-3)

Since the gradients are proportional to the velocities we can rewrite

Equation ( 3-3) as

FP, .(P 2 ,,f';,)= f vP, b (P,r;r') (3-4)

Now both sides are proportional to the magnitudes of both F1 and F2 .

We can then choose both to be of magnitude 1. We choose -P and A 2

separately to be 1 of 3 orthogonal unit vectors el, e2, e3 . This yields

the relations

ý&j(ei, f'; r') "- ýj(ii,r, f ' (3 -- 5)

(i = 1,2,3).

Thus the velocity is the ith direction at r, due to a force in the ith

direction at i' is the same as the velocity in the ith direction at r. due

to a force in the ith direction at ,'1 .

If we choose P, and P2 orthogonal, we get the 6 relations

ý (Zj, f', , ) = iiA(,,,'; i) (3-6)

8

i j.

Thus the velocity in the ith direction at f' due to a force in the jth

direction at €2 is equal to the velocity in the jth direction at F' due to

a force in the ith direction at fl'.

c) LetF1 = 0, M1 = Mb(F'- f1 ),

F2 = P26( -r 2 ), M2 0.

Then P&(i') = P(FP = 0, M, f 2; f')

P2 = P(F2,0, F2

Equation (2-7) becomes (on replacing pressure gradients by velocities)

iwMP(P2,0, r';iF) =- F2 "( =,iwpo

Since both sides are - M and (P 2), we can take M = 1, and P 2 = e, where i

- 1, 2, 3 yields 3 orthogonal unit vectors.

Here we have the three relations

w2poP(li,0,r';f1)=vi(FI = 0,1,ý,;if) i= 1,2,3 (3-7)

In other words this says: The pressure at i, due to a force at F' in the ji

direction is proportional to the velocity in the ith direction due to a source

at r, at F2.

We hazard the guess that this is the reciprocity principle referred to in

reference [1].

9

References

[1] Donskoy D., 1991 "Soviet R&D in Low-Frequency Underwater Acous-

tics," published by Delphic Associates Inc., 7700 Leesburg Pike, No.

250, Falls Church, Va 22043 .

[2] Rayleigh, J. W. S., 1945, "The Theory of Sound", Vol II, p. 145, Dover

Publications, NY.

[3) Landau, L. D. and E. M. Lifshitz, 1959, "Fluid Mechanics," Pergamon

Press.

a) No. 74

b) Problem on p.291

[4] Morse, P. M. and K. U. Ingrad,(1986), "Theoretical Acoustics", Prince-

ton University Press, Princeton, NJ, p. 134 , 312, 320, 342, 405 (1986).

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