DTIC ADA261515: Structural Acoustics: A General Form of Reciprocity Principles in Acoustics

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SiructUralAcousticcs:
A General' Form of Reciprocity
in Acoustics
"Priciple~s
DIC
93-04246
MTRE
Reproduced From
34 W Best Available Copy
Structural Acoustics:
A General Form of Reciprocity
Principles in Acoustics
K. Case
Accesion For
NTIS CRA&I
DTIC TAB
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Avail andI o
Januar 1993
JSR-92-193
Approved for pubbe release dltbutobn un tmed.
JASON
The MITRE Corporation
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REPORT DOCUMENTATION PAGE
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Ilanuary 1,1993
1. "iEn" USt DAlY (Le"V. Wik .IR EPORT DATE L. 1 TP M PIR 6WCW
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4. T AND SUBTMEITL .... 1, 3 K
Structural Acoustics: A General Form of Reciprocity Principles
in Acoustics
S. AUTHOR(S) PR - 8503Z
K. Case
7. PRPORMI(cid:127) G ORGANIZATION NAME(S) AND AOORISS(ES) 1. PiRINOti OIGANIZAT)ON
The MITRE Corporation
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11. SUPPLEMENTARY NOTES
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Approcved for public release. Distribution unlimited.
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.P IKE coot
17. sECUMRT CLASSCAT(cid:127)1(cid:127)I, SECURTY CLASSUICATION It. SECURIrY CLASSWCARI 20. UMAHTArN Of ABSTRACT
OP REPORT OF THIS PAGE OP ABISTRACT
UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED SAR
NSN 7140Q-1-290-S500 Stamdard Form 296 (Rev 2-69)
Ž6rot 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
ap
5t+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)
Poj = -VP
7
op
a-+poV-v
= 0.
N--" = '9" The resulting equations are
Po-: = -VP
3
1 oP + P0 v . 0.
2- t6ý--
Assuming simple harmonic motion, i.e., all quantities - e-iw' yields.
-iwpot3 = -VP
C2 -p+ po V 6 =. 0.
Let us introduce forcing terms into both equations. Then
- iwpo0 = -VP + P(i - f ) (2-1)
1
-iWP + poV .I = M(f - ý ). (2-2)
2
Eliminating 5 we obtain
(V2 + k2)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
F , M ). Thus, we have
2 2
(V'2 + k2)P1(F') = V' F1(i' - (cid:127)) + iwM1(i' - i) (2-4)
(V' + k2)P2(,') = V' .(cid:127) 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:
Jd
3i (P2( I')V* P&') - P-I-)V'2P2fP)}
4
= d3r' {P,(F')V'.(' - I) + iWP2(f')MI(i:' - r)} (2-6)
fd3r' {Pi(P')V'. ) + iwP&(2')M (' r )}
- -( 2 - 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'{P (P')V'. • (?' - f') + iwP (ý')Mi(F' - ri)}
2 2
= ,d{3IrP' (F')V'.• (F' - (cid:127))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'-((cid:127) i ) equal to F16(f - ') (PA now
a constant vector) and M (- - i1) equal to Mb(9 will be
1 - 72)
denoted as
a) Let P, = =O.
2
M, (f' f,) --- M6(f,'- fa
M2(f -r2) M~' 2
(M a constant).
Then Equation (2-7) reads
P(P"= 0MU,l ,; 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 ) = P(f2; ý1). (3-2)
2
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