NBSIR 73-329

THEORY OF ADlOiNT RECIPROCITY FOR

ELECTROACOUSTIC TRANDUCERS

A.D. Yaghjian

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

February 1974

NBSIR 73-329

THEORY OF ADJOINT RECIPROCITY FOR

ELECTROACOUSTIC TRANDUCERS

A.D. Yaghjian

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

February 1974

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS. Richard W Roberts. Director

CONTENTS

Page

INTRODUCTION 2

I. THE TRANSDUCER SYSTEM 6

II. RELATIONSHIP BETWEEN THE TWO FORMULATIONS 12

A. Derivation of the Scattering-Matrix Equations from the Spatial Impedance -Matrix

Relations 12

B. Equivalence of the Alternate Forms of

Reciprocity 19

III. ADJOINT RECIPROCITY 24

A. The Operator Matrix Description 25

B. The Adjoint Operator Matrix and an Electroacoustic Reciprocity Lemma 32

C. Generalized Reciprocity Relations 39

D. Adjoint Reciprocity Theorems 46

E. Power Relations 52

APPENDICES 58

ACKNOWLEDGMENT 68

REFERENCES 69

iii

THEORY OF ADJOINT RECIPROCITY FOR

ELECTROACOUSTIC TRANSDUCERS

A. D. Yaghjian

ABSTRACT

Analytical techniques for the measurement

of the external characteristics of electroacous tic

transducers have been developed by D. M. Kerns using

a plane-wave scattering-matrix (PWSM) formulat ion

.

Foldy and Primakoff, in their classic papers on

linear electroacous tic transducers, utilize a spatial

impedance -matrix (SIM) formulation. Both formulations involve a continuous, linear "matrix" transformation in which reciprocity is defined as a relationship between elements of the matrix.

The first portion of the present report demonstrates that a transducer satisfying the "SIM relations" also satisfies the "PWSM equations" (but

that the converse theorem does not hold), and that

the alternate expressions of reciprocity are equivalent for transducers that obey both formulations.

The second portion of the report examines the

equations which characterize the internal behavior

of a large class of elect roacous tic transducers.

A linear operator approach is employed to develop a generalized reciprocity lemma which is used to

establish adjoint reciprocity relations between the

fields of a given transducer and its adjoint transducers. The linear operator approach facilitates

the identification of self-adjoint (reciprocal or

ant ireciprocal ) transducers, and the adjoint

reciprocity relations have utility in the extrapolation techniques of the PWSM formulation.

An adjoint "reciprocity theorem" and "principle

of reciprocity" are derived from the generalized

reciprocity relations. Finally it is shown that

the total power inputs for the adjoint transducers

belong to the same "value class" as the original

transducer.

Key words: Adjoint operators; e lect roacous tic transducers; reciprocity; scattering matrices.
INTRODUCTION

12 3 Analytical techniques developed by Kerns * * for the

measurement of microwave antennas have been "translated"

recently by that author into corresponding techniques for

the measurement of electroacous tic transducers in fluids.

The basic theory is formulated with a scattering-matrix

description and emphasizes the use of plane -wave spectra

for the representation of acoustic fields. For a summary

and for details of the theory and its variety of applications,

4

reference should be made to Unpublished Report.

Although the electromagnetic (em) theory is quite

comprehensive , the acoustics applications are greatly

strengthened by generalized or adjoint reciprocity relations,

the derivation of which forms the principal subject of the

present report. In addition, the relationship between the

classical spatial impedance -matrix (SIM) description of

electroacous tic transducers used by Foldy and Primakoff

,

and the plane-wave scattering-matrix (PWSM) description

introduced by Kerns, including the expression of generalized

reciprocity in each scheme, is investigated.

Kerns begins the PWSM description by expanding the

external acoustic field in a double Fourier transform

representing a continuous spectrum of plane -wave modes.

He then relates the amplitudes of the outgoing acoustic waves

2
b(K) and the emergent em mode to the amplitudes of the

incoming acoustic waves a(K) and the incident em mode a^

by the scattering equations,

= S^^a^ ^ / S^-L(L)a(L) dL (la)

L

b(K) = S^^(K)a^ + / S^^(K,L)a(L) dL, (ib)

L

in which the integrations extend over the infinite I I ^ X y

plane. r The vectors K = k i + k j and L = i i + i ] may be

X y-^ X y-^ ^ interpreted as the "transverse" parts of the plane-wave

propagation vectors k and I. Equations 1 resolve the combined

scattering matrix into four submatrices S t , St , S,, , S

^ ol ' lo ' 11 ' 00

which represent the receiving, radiating, acoustic scattering,

and em reflection properties respectively for the electroacoustic transducer.

7 In analogy to the 2-port cases discussed by McMillan,

4 Kerns defines reciprocity, ant ireciprocity , and nonrec iproci ty

for electroacous tic transducers in terms of whether the following relationships hold with the plus sign, the minus

sign, or not at all:

n^S^-^CK) = ± n(K)S^^(-K) (2a)

n(K)S^^(K,L) = n(L)S-^^(-L,-K) . (2b)

The quantities and n are admittances which depend upon

conventions of sign and normalization.
The classic papers of Foldy and Primakof £^ * ^ use a spatial

impedance -mat rix formulation to characterize the external

operation of linear electroacous tic transducers. The SIM

g relations may be written,

V = Z, I + / h' (r )u (r ) da (3si) o b o ^ ^ o^ n^ o^ o

^-^"-J

p(r) = h(r)I^ -H / ZJr,r^)u^(r^) da^ , (3b)

o where the four spatial impedances Z^y h', h, and Z^ characterize a transducer by relating the "voltage"

V = a + b C4a") o o O

K^°-J

and excess pressure p on the surface of the transducer

(except possibly on the feed area S^) to the "current"

I = (a - b )n r4bl

o o o-^ O ^^^J

and inward (to transducer) normal velocity u^ on that surface.

Although three of the four spatial impedances depend upon

the position vectors (r,r^) to the surface A^ and all four

depend upon the harmonic time frequency, they are assumed

independent of the medium or sources surrounding the transducer and, as linearity implies, independent of V^, I^, p

and u . n

A transducer satisfying the SIM relations 3 is defined

as reciprocal by Foldy and Primakoff if the following relations

4
among the spatial impedances are satisfied:

h' (r) = ± h(r) (5a)

Z (r ,r ) = Z (r ,r) . (5b)

Otherwise the transducer is nonreciprocal . They do not use

the term "ant ireciprocity" but distinguish between the plus

and minus sign in Eq. 5a by considering reciprocity through

electric -type coupling and magnet ic -type coupling.

The first portion of the present paper demonstrates that

the SIM relations 3 imply the PWSM Eqs . 1 (but that the

converse theorem does not hold) , and proves that the alternate

expressions of reciprocity, Eqs. 2 and 5, are equivalent

for transducers which obey both the SIM and the PWSM formulation.

The second portion of the paper examines the equations

which characterize the internal properties of a large class of

elect roacoust ic transducers. A linear operator approach is

employed to develop a generalized electroacous tic reciprocity

lemma which relates the field properties of a given transducer

to those of its mathematical adjoint. The linear operator

approach also facilitates the identification of self-adjoint

transducers, transducers which satisfy the usual reciprocity

relations 2 or 5.

5
The reciprocity lemma is used to establish adjoint

reciprocity relations, which have utility in the "extrapolation techniques" of the PWSM formulation.^ (Extrapolation

Q techniques were introduced for antennas by Wacker and

Bowman. The method is applied and briefly described in

a paper by Newell and Kerns. ^^) The "electroacous tic

reciprocity theorem" and the "principle of reciprocity"

for scatterers are extended to mutually adjoint transducers

through use of the adjoint reciprocity relations.

Finally, power relations and the associated "value

classes" are investigated for electroacous tic transducers

and their adjoints. In particular, it is shown that the

power input to the adjoint transducers belong to the same

value class, i.e. have the same "definiteness" as the power

input to the original transducer.

I. THE TRANSDUCER SYSTEM

The electroacous tic transducer system under

consideration is pictured in Fig. 1. The transducer is bounded

by the closed, finite surface . The em source or detector,

which is bounded by the closed, finite surface A^ , feeds

the transducer through the area S common to both A and A . ^ o op

The transducers, which may contain static bias fields and

their sources, are termed "passive" if they cannot radiate

6
Fig. 1. Schematic of the transducer system.

A=Aq+Ap denotes the external surface of the

electroacoust ic transducer-plus-source or detector.

7
more power than they absorb. The present analysis, however,

is not limited to passive transducers, but may include transducers which are active because they contain "gainy" material,

or radiating sources, or both. (A transducer will be called

source-free i£ its internal equations of motion are homogeneous, i.e., if they are satisfied by all fields equal to

zero .

)

The transducer-plus - source (or detector) is immersed in

a homogeneous, isotropic, stationary fluid which extends to

infinity and supports the time harmonic (e -^^^^ acoustic

pressure -velocity field of small (first order differential)

amplitude. The static pressure, static density, and propagation constant (the latter of which may be a complex function

of real frequency 03 to account for viscous and "expansive

friction" losses) are constants in space and time denoted by

, and k respectively. The fluid between the infinite xy

plane (z=0) and the external surface A of the transducerplus-source contains no acoustic s ources , although

arbitrary sources (at frequency 00) may exist to the right

of S-^.

The normal velocity on the surface Ap of the source

detector, including the feed area , is assumed negligible.

Electromagnetic shielding necessary for deriving the adjoint

reciprocity relations is discussed in Section III.C.

8
The area may designate the perpendicular cross

section o£ an open or closed em waveguide (uniform and isotropic) in which a single em mode is propagating with incident

amplitude a^ and emergent amplitude b^, or simply the cross

section of two wire leads when the frequency o) is low enough

to allow analysis by quasi-static voltages and currents. The

cross -sect ional area for a mode on an open waveguide (such

as a Lecher line or microstrip) must extend sufficiently far

from the guide, to insure that the fields in the mode outside

contribute negligibly to the normalization integral (Eq.

11 below). The surfaces , , and represent imaginary

boundaries in that they are chosen to facilitate the theoretical analysis and need not coincide with the physical

boundaries of the system. Of course, the surface A = A^ + A^

must be chosen within (or on the interior boundary of) the

ambient fluid.

The continuity and momentum equations, which govern

the harmonic acoustic fields in the source-free fluid between

the surface A and the infinite plane , may be expressed as

k^p = -iojp^V'U (6a)

Vp = ioop^u, (6b)

where the "small," complex amplitudes, p and u, denote the

position-dependent excess pressure and fluid velocity. The

effect of gravity on harmonic variation in mass density is

neglected

.

9
The excess pressure and normal velocity on the surface

may be expanded in a double Fourier integral (transform)

over plane-wave "voltage" and "current" spectral densities:'

p(R) =

2tt V(K)e^^'^ dK

K

u fR) = - ^

I(K)e^^*^ dK.

(7a)

(7b)

K

Conversely, the plane-wave voltage and current densities,

V(K) = a(K) + b(K) (8a)

I(K) = [a(K) - b(K)]n(K) , (8b)

may be expanded in a double Fourier integral (inverse transform) over the excess pressure and normal velocity:

V(K) = ^

27T p(R)e dR (9a)

I(K) =

271

fjr^ -iK'R u^(R)e dR. (9b)

The vector R is confined to the xy plane S-j^, and the z-compo

nent of the plane-wave admittance ri(K) equals y/'^Pq* where y

is defined as (k^ - K^) ^ with the sign of the radical chosen

to keep the real and imaginary parts of y positive.

The transverse em fields (E^ , H^) on the cross

section of a waveguide may be written as the sum of the

10
transverse "electric" e^ and "magnetic" components of the

mode traveling to the right (incident) and left (emergent)

:

E

t

= Vo' "t = ^0^0' CIO)

where V and I are defined by Eqs. 4.

0 0 / n (Note that here the terms "transverse electric and magnetic"

are not used to denote TE and TM modes.)

If the dimensions of e" and PT are chosen as (meter) ^ 0 0 ^ ^ and (Sq, iIq^^) are consistent with Maxwell's equations,

behaves dimens ionally as volts, as amperes, and as an

admittance (ohm "*") normally chosen as a positive real number.

Moreover, normalization may be expressed as a nondimensional

number equal to unity, i.e.

/ (e^xh^) -fi da = 1, (11)

s

o

where n is the inward normal to the transducer. (The rationalized MKS system of units is used throughout.) If the area

simply cuts two wire leads at quasi-static frequencies,

and I^ refer to conventional circuit voltages and currents

which do not necessarily serve as genuine modal coefficients.

In that case, Eqs. 4 become a definition of a^ and b^.

11
II . RELATIONSHIP BETWEEN THE TWO FORMULATIONS

This section shows that the scattering -mat rix equations

1 can be derived from the SIM relations 3 (but not conversely) , and that the two expressions o£ reciprocity, Eqs

.

2 and 5, are equivalent in that each expression implies

the other for transducers that obey both descriptions. These

results are established without reference to the internal

properties or behavior of the electroacous t ic transducer.

A . Derivation of the Scattering-Matrix Equations

from the Spatial Impedance -Mat rix Relations

Application of Green's theorem to the excess pressure

(henceforth referred to as just pressure) in the fluid region

between the external surface A of the transducer -plus -source

or detector and the hemisphere closed on the right by the

infinite plane S^ , produces

p(r) = 9G(r,rQ)

P(r^) —- da^ - ia3p^

dn o o S^

(12)

u (R )G(r,R ) dR .

The pressure and Green's function in Eq. 12 satisfy the

scalar wave equations,

V^p + k^p = 0 (13a)

G(F,F^) + G(F,Fq) = 6(F-F^), (13b)

12
where 6rr-r ) denotes the three-dimensional delta function,

o The boundary conditions for the Green's function have been

chosen as

8G(?,? ) =0 r on and A -S ri4al

8n

o 1 p o ^

J o

G(F,F^) =0 on A^-S^. (14b)

Also, by means of Green's theorem, the Green's function with

the prescribed boundary conditions can be shown to possess

the symmetry property,

G(F,F^) = G(F^,F). (14c)

The exponential decay in lossy fluids or the radiation condition in lossless fluids insure the vanishing of the integrals

over the hemisphere at infinite radius. Equation 6b has been

used to relate the normal derivative of pressure on A and

S-^ to the normal velocity there, i.e.

/i.Vp = IP = icop^u^ (n = e^ on S^) (15)

9n

where the unit normals are directed away from the region between A and S^. (The existence of unique mathematical solutions for p, u, and G, here and in the ensuing expressions,

has been assumed implicitly.)

Substitution of pressure from Eq. 3b and normal velocity

from Eq. 7b into their respective integrals in Eq. 12 yields

13
_ _ 9G(r,r ) ioop iK-R

p(r) = / h(r^) da^ + ^ / / I(K)G(F,R^)e ° dR dR

A 2TT 3^

(16)

9G(r,r^)

+ /<<o^o'l^^ / Z (r yV^) u (r.) da da, (r on A). n^l-^ o 1 ^ ^ A A dn

o After differentiation with respect to the normal direction on

A and utilization of Eq. 15, Eq. 16 recasts into an integral

equation determining u^ on the surface A in terms of the

"currents" and I (K)

,

u^(F) =

/ DC?,?^) u^(F^) da^ + g(F) (? on A). (17)

A. The kernel is given by

1 _ _ 9'G(F,F ) D(r,r^) =

/ Z^(r^,r^) -—- da^ , (18a)

leap A 9n9nT

o 1

and the source function by

I _ 92G(r,r ) g(r) =

/ h(r^) da^

loop A 9n9n

•^o o

o

(18b)

, _ 9G(r,R ) iK-R

+ — / / I(K) — ° e ° dK dR

27T g ^

9n

Because u vanishes when I and I (K) vanish, nontrivial

n o ^ ^

' homogeneous solutions to this two-dimensional Fredholm integral

equation 17 of the second kind do not exist. Thus, a unique

14
mo

t/5

C/) CD CD 1 r~l •rH5h c;

cxu CD 0

t-H IT) •H

rH ,—(

(-> vU w rH u • ri rH Cl H 4-* TO O U f 1 1 HH

p—(

>—> CD

H

Tj t [ 1 Co

CD CD

C/1 f \ 1 rH rH •H

(D

t,M rH

5. r-* p-> 'rH PH CJ r> Co X If) CD

r^

(1 1rrt to • rH CO rHP

CD

r-l r^

CJ c •rH CO <D M

)-> •rH TO 1 H C/) M rH

PI *rH to •rH ' rH rH Mh

r-< 00 CD

<U

rH

i-t r^

H

H 1 rH 1 1 CD rH

(tI 1 H

U (_, M

CD (IN rH >> n-H rH PH • H iHCo Cj

CD »<:L, Co

f-N. + CD

r-l r—1

00

frt rH rH ! rH TO

rH Ph cr

pH CjiJ

XW /1 1

CD

W \>

II r-< u;

t/) ( 1 J MH

t—> rH •H CJ

4—* vU 1 S_ U +-> Tj 0 • H •rH H rH u Oj

P-t CD 0

•rH I—1

CD

U 5-H CD rP

r-i CD CD > M H

I—1rP

M P cti -P

•rH CD ?H :s 0 M >

4-> cl rH

CD

c! > CD 0 rH CD H '£ fH M 1—1

;3 CD CD rH A

03

03

T3

03 j=i o o

1?0

rH d

CN] CD

CD

CD

|5h

rC

Q.3

•H

03 t3 0

<M CD

CD c; CD

1^

CO. 3

1^

iH

03

•T3 I—

v

V /

CD

cl CD

•H

CD

CD

CD

(Nl o

|!h

|!h

1^

Ou

(D

0)uouowmo

•H

t/5

•(->

•H

•P>^

(DU

03 rHPh

(DfH

(Ud

03U

cr w

ci •H

•P •Huo

rH

CD>

03

fHo

CD H

15
1—\ /—\

rQ

CVJ I—1

CNl CNl (NI \ 1 •T3 o

Id

1^<I—\ +I—

1^

PL,o

0)

•(-> cd

r-H

5h

CDU<

03 c <

0)5h

CD

IS

03

03 t3

1^

1^< '

CD

1^

o3

•T3

1^

rd

OJ

1^N I

1^

ISI . 0/—

c o

Q.3• H

.CO

(Nl

II N

<i5

tfl

5-1 >

•H0

•P

•H

o3o

"-H

t/l 4-> rH

;=s

?-i

c!o • • •H I—1

•P CO

03

;3 (D

c!

D o3 I—1 X

•H

}h 0) P

03 P S1 ci

(D o U

cl CM

03 (Nl• cr w

•H o

1—1

rH e

03 ?-< u o p m U

(U

!=U 03

t/5 P

CDa 5h o CD •H

?H

^3O

\^

T3

•H

CD I—\ </—v

r- . CO

t=

CNl

l»5 T3

CD

cn

O t=

H (Nl

t3

•H

CD

CO

p-ll CNl

pr1^> 03

CM

10^

CD

/^

—\

.CO

CNI

rrISI >N o

1—1

'£ CD> o •H \^ 1—1 bO

1—1

II I— •H1 CD o

ISI • .CO

C/) 1 t= CD 1—1 1 (Nl U

II 03 nd

CD t—1

IN3

•H

(D rdPP

•H

IS

16
Similarly, the second spectral impedance -mat rix equation

is determined by substituting the normal velocity of Eq. 21

into the SIM relation 3a, which becomes

= ^00^0 / Z^i(L)I(L) dL, (26)

L

when the impedances Z and Z ^ are defined as ^ oo ol

^oo = h ^ ! h'(F)F^(F) da (27a)

A.

Z (K) = / h'(F)F(K,F) da. (27b)

A

The spectral mat rix- impedance representation (Eqs. 24b

and 26) combine with the definitions of the generalized

voltages and currents, Eqs. 4 and 8, to form the scattering-matrix equations 1. (The derivation of S^^ , , S-j^^,

and S, T from Z , Z ^ , and Z^^ is a straightforward,

11 00 ' ol ' lo 11 ^ ' reversible, but lengthy procedure which parallels the analagous derivation for 2-ports.)

In brief, every electroacoust ic transducer which satisfies

the SIM relations can be described by a spectral impedance or

scattering matrix. Moreover, the elements of these matrices

are determinable as explicit functions of the four spatial

impedances (Z^, h*, h, Z^) that characterize the transducer.

The following investigation shows by means of a simple

class of counter-examples, that the converse theorem does

not hold.

17
Consider an elect roacous tic transducer with I held zero

o but subjected to an incident acoustic field from without.

Suppose also that the transducer behaves as an acoustically

rigid (u = 0) scatterer when I is zero, and that V is

n o o linearly related to the pressure on A. The SIM relations 3

do not describe such a transducer because p and V do not ^ o vanish even though u and I do. ^ n o The same rigid transducer with zero and linearly

related to pressure on A is described, however, by a scattering matrix. Integral equation 12 can be solved for pressure

on A as a function of the normal velocity u^ , and the result

substituted back into the first integral in Eq . 12. If

the normal velocity is then expressed as the integral in

Eq. 7b over I (K) , and the inverse Fourier transform in

the S-j^ plane is taken of the resulting equation, the impedancematrix equations corresponding to Eqs. 24b and 26 with

zero I^ appear. The impedance -matrix equations in turn convert into the scattering-matrix equations 1 with a =b . ^ ^ 0 0

Rigid scatterers may not represent the only examples of

linear transducers which can not be described by the SIM

relations, but they suffice to demonstrate that the PWSM

description applies to a larger class of transducers than do

the SIM relations.

18
B . Equivalence of the Alternate Forms of Reciprocity

Consider the separate definitions of reciprocity expressed by Eqs . 2 and 5. Although it was just shown that

the PWSM equations 1 describe a larger class of transducers

than the SIM relations 3, the alternate expressions for

reciprocity prove equivalent for transducers which obey both

descriptions. Equivalence is established by deriving the

PWSM reciprocity relations 2 from the SIM reciprocity relations 5, and vice versa.

The spectral impedance matrices , ^lo' ^11 '^^^

be expressed through Eqs. 27b, 25, 23, and 20 as functions

of the spatial impedances h, h', Z^, and the Green's function,

provided the resolvent kernel P(r, r^) in Eq. 19 is determined

as a function of the same spatial impedances.

Successive substitution with the source function in

Eq. 17 determines the solution for the resolvent kernel as

12 the Liouville -Neumann series,

V(r,T^) = D(r,F^)+ / DCF,7^)D(7^,F^) da^

+ / / D(r,r2)D(r2 ,r^)D(r^ ,r^) da.^ da^

+ (28)

In spite of its strong requirements for convergence, the

Liouville -Neumann series is used in Eq. 28 instead of the Fredholm series, which converges except at the eigenvalues,

19
because identical results are obtained using either series,

but the mathematical details are shortened considerably.

Insertion of the original kernel D of Eq. 18a into the

expression 28, Eq. 28 into Eqs . 20, Eqs . 20 into Eqs . 23,

and Eqs. 23 into the impedances 25 and 27b, results, after

some rearrangement, in the final expansions for , Z^^,

and Z^-j^ :

, 3G(r ,R) .:rr tT °1 2TT L A ° 9n

1 o

27Tiojp S, A A A an.

u

^ ± o 1 1 o 2

e^^*^ da da, da^ dR + (29a)

o 1 2 ^ '

, 8G(R,r ) .iy^.n _

Z (K) = ^

/ / h(r ) ° e ^ da dR

2tt S, a 8n

3GCR,F ) d^G(T

- —— ! ! I ! h(F^) — ' ^o^^i'^i^

2Tricop St A A A da.

u x ^ 9n^8n

^o 1 1 2 o

e"^^*^ da da, da^ dR + ... (29b)

o 1 2

^

20
^^Pn iW V — iL • R

Z^.CK,L) = ~ ! ! e"^^*^G(R,R^)e ° dR dR

1 r r r r -iK-R - - (R,? ) 9G (F R )

(27T)2 S-^ A A ° ^ ^ dn^

iL-R^

e da^ da^ dR^ dR + (29c)

Only the first two terms in the expansions 29 are carried

explicitly, but the essential "symmetry" property required to

prove Eqs . 30 beloxv is found in every term.

The symmetrical properties of the Green's function

GCr,r^} and the impedance Z^(r,r^), as expressed by Eqs. 14c

and 5b respectively, imply from Eq. 29c that

Z^^(K,L) = Z^^C-L,-K) . (30a)

Similarly, Eqs. 14c and 5 imply from Eqs. 29a and 29b that

^01^^^ = - ^10^-^^- C30b)

Again, by a straightforward, reversible, but lengthy

exercise which will not be included here, the impedance

reciprocity relations 30 combine v;ith the definitions 4

and 8 and the scatter ing—mat rix equations to form the PWSM

reciprocity relations 2. The ± signs in Eqs. 2a and 30b

are coupled with the ones in Eq. 5a.

The completion of the proof of equivalence consists in

deriving the SIM reciprocity relations 5 from the PWSM relations 2.

21
If the Eqs . 7 for pressure and normal velocity, the

spectral impedance Eq. 24b, and the SIM Eq. 3b are inserted

14 into the acoustical reciprocity theorem,

as applied to any two acoustic fields supported by the fluid

between A and S^, the following expression emerges:

/ / u-'-(r )[Z (r,r ) - Z (r ,r)]u^(r) da da n^o^'-o^'o^ o^o'-^-'n^'^ o

+ / h(F) [lV(F) - I^u^(7)] da

=

/ / I^(L) [Z-^^(K,L} - Z-^^C-L,-K)]I^(-K) dK dL

L K

H

/ Z^J-K) [I^I^CK) - I^I^K)] dK. ^32)

K

Again, the exponential decay in lossy fluids or the radiation

condition in lossless fluids insure the vanishing of the

integral in Eq. 31 over the hemisphere at infinite radius.

The scattering reciprocity relations 2 imply the

spectral impedance reciprocity relations 30, which in turn

imply the SIM reciprocity relation 5b,

by letting

Z (r ,r ) = Z (r ,r) ,

- - 0 o o

22
in Eq . 32

The derivation of Eq. 5a from Eqs . 2 requires more

effort. Since the first integral on both sides of Eq. 32

is zero and Z^^(-K) = ± Z^.^^ (K) , Eq. 32 may be written as

/ h(F)u^CF) da i

/ 1^^(1)1^ m dK = 0, (33) ^ K

2 2 —

if is chosen zero. The additional choice of I (K) =

6(K-K^) and use of Eqs. 21 and 27b transforms Eq. 33 into

the two-dimensional homogeneous Fredholm integral equation

of the first kind,

/ [h(r) + h'(r)]FCK,r) da = 0. (34)

A

Under the assumption that the kernel F(K,r) does not have zero

as an e igenvalue , the one continuous solution to Eq. 34 is

h(F) = ± h'(F), (35)

which completes the exercise to demonstrate the equivalence of

the PWSM and SIM reciprocity relations.

Although the reciprocal symmetry of the spectral impedance and scattering matrices has been derived from Eqs. 29

and the symmetry of the spatial impedances, an alternate approach can be taken from the acoustical reciprocity theorem

32. Such an approach, however, encounters mathematical

difficulties which discourage its use.

23
III. ADJOINT RECIPROCITY

For reciprocal transducers, the transmitting and receiving characteristics determine each other, the speaker and

microphone response functions become proportional,^ and

scattering in one set o£ directions relates directly to

scattering in the opposite set of directions. In general,

the analysis and measurement needed to characterize an

elect roacous tic transducer are reduced if the transducer is

known to obey the reciprocity relations. Moreover, some

techniques are applicable to reciprocal transducers only.

Kerns has shown, for example, that the transmitting and receiving characteristics of two unknown but identical transducers obeying reciprocity and possessing symmetry about an

axial plane can be determined by a single transverse scan of

4

one transducer by the other.

In addition to its direct importance in simplifying

transducer analysis and measurement, reciprocity is used in

the plane-wave scattering-matrix theory to express mathematically the receiving matrix S^-j^ as a function of

1

7

equivalent sources within the transducer. Such expressions

have become especially important because of their use in the

theoretical foundation of extrapolation techniques which predict gain on an arbitrary axis from measurements at reduced

9 10 distances along that axis. '

24
For nonreciprocal transducers, the expressions for S^-j^

in terms of equivalent sources no longer apply unless there

exists an operator or transducer (the adjoint transducer) with

transmitting characteristics S^^ related "reciprocally" to

the receiving characteristics S^^ of the original transducer.

The remainder of the present report investigates the internal

operation of a general class of linear electroacous tic transducers to prove the mathematical existence of the adjoint

transducer and to determine its precise relationship to the

original transducer.

Specifically, the equations describing the material

properties and behavior of the transducer are written as a linear operator matrix. The operator is "transposed" to form

the adjoint operator, which is compared and combined with the

original operator to derive a generalized electroacous tic

reciprocity lemma. The reciprocity lemma leads to the generalized reciprocal relations between the characteristic "matrices"

of the mutually adjoint transducers. A similar analysis for

1

8

antennas has been performed by Kerns.

A. The Operator Matrix Description

Primakoff and Foldy, in their original work on electroacoustic reciprocity, examined the equations which model the

internal operation of a large class of transducers. They

found that if certain "symmetry conditions" were satisfied

25
by its material parameters and static fields, the transducers

obeyed the SIM reciprocity relations. The present derivation

begins in a similar way by writing the linear, harmonic

(e ^'^^) equations that govern the "physics" of the material

within the transducer A : o V-T + ia)p° u + r = 0 (36) .

VxE - iojB = 0 (37a)

VxH + iwD - J = 0 . (37b)

The momentum equation 36 expresses Newton's second law for

stationary (u^ = 0) media, and Eqs. 37 express Maxwell's

equations in differential form for stationary media in which

regions of electric and magnetic polarization may exist.

The static quantities, which will be distinguished by

the superscript or subscript "o," are real functions of the

space coordinates. All other quantities, except the real

frequency oo, represent complex, space -dependent , harmonic

amplitudes, which superimpose upon the static variables as

products wi th e'^"^^. For example, if the total stress dyadic

at a point in space and time is denoted by T'(r,t) then T'

can be divided into a real static part T^(r) and a complex

harmonic part T(r)e ^^'^

^ i.e.

T' (F,t) = T^(F) + T(?)e"^"^^

Of course, it is the real part of T' that corresponds to a measurable quantity. Only the complex harmonic amplitude T(r) shows

in Eq. 36.

26
Similarly the total mass density p^(r,t) may be divided as

p'(r,t) = p°Cr) + p (r)e'^'^^,

but only the real static part occurs in Eq. 36. In fact,

the mass continuity equation is not included in addition to

the momentum equation 36 because it merely introduces the

extra variable p (r)

.

The other variables in Eq. 36 are u, the harmonic

velocity of the material, and 7, the total harmonic volume

force. The usual harmonic em field vectors are denoted by

E, H, B, D, and J. All variables refer to "macroscopic"

quantities which are evaluated, at least in concept, by

averaging "microscopic" variables over "small" but finite

volumes and time intervals.

The number of unknowns in Eqs . 36 and 37 reduces to

the number of equations if the following linear constitutive

relations characterize the transducer material:

D=e*E+T«H+a:s+P (38)

(39)

(40) = t =1

T = c:s - a'E - 3'H (41)

£ = p° E . J X B^ . T3, (42)

27
where the strain dyadic s is related to the velocity by

s = - — I^Vu - |(Vxu) X I ICO (43)

(I is the identity dyadic)

The various "dot products" involved in the vector and dyadic

transformations of Eqs . 38-43 can be understood in terms of

Cartesian tensor notation, in which e*E, a'*E, a:s and c:s

become e. .E . , a^.^E^, a..^s.^, and c-.j^^s^^ respectively

(summation over repeated dummy indices is implied) . The

19 "n-adic" notation is used wherever convenient because it

preserves the familiar vector- dyadic notation for the electroacoustic fields. ^'^^

The dyadics a, e, y, and (t,v) stand for the conductivity,

2

1

permittivity, permeability, and magneto-electric properties

of the transducer material.

The direct piezoelectric and pie zomagne t ic stress triadics,^^ a and 3, may be chosen symmetric:

a. = a. 1

. Ilk iKi

ikj ,

(44a)

(44b)

since the strain dyadic is symmetric.

2 ioo

8u.

9x:

9u,

dx (45)

Similarly, because the stress is a symmetric dyadic (Tj^j = ^j^)

(provided there are no distributions of body torque, stress

2 3 couples, or angular momentum density), the converse piezo28
electric and piezomagnet ic stress triadics, a' and 3', must

possess the symmetry,

3-jk = e^j^. (46b)

The symmetry of the stress and strain dyadics also

requires the Hooke's tetradic c to obey the symmetry

''ijkil ' ^jik£' C47a)

and allows the defined symmetry

^ijk5, = ^ijilk- (47b)

The static charge distribution p° , the static magnetic induetion B ^ , all other static quantities, and material parameters

which may depend upon frequency w, remain independent of the

values of the harmonic variables.

Although only transducers which obey the homogeneous

equations are considered finally, the harmonic sources

of polarization , magnetization (y^ is the permeability

of free-space), current J^ , and volume force have been

included in Eqs. 38, 39, 40, and 42 to allow insight into

applications involving these sources.

A number of other assumptions underlie the development

of Eqs. 38-43. The harmonic displacements (rotation as well

as strain) about the static positions must be small - actually,

differentials of first order - to insure the validity of the

strain-velocity relationship 43. Consequently,

» 29
all the harmonic field variables are assumed to be first

order differentials.

In the current equation 40 the "Hall effect" of the magnetic induction on "drift" velocity is ignored, as well as harmonic variations in conductivity which might be found,

for example, in a biased carbon microphone. A carbon microphone would not be described by the constitutive equation 40.

The volume forces on electric and magnetic dipoles

,

static currents, and harmonic variations in mass density

(acted upon by gravity) have been omitted in the force Eq. 42.

The effects of gravity are usually minute, and if the material

that contains the dipoles or static currents is rigid or has

insignificant effect on the essential behavior of the transducer, the volume forces on the dipoles and static currents

are justifiably neglected also. The magnet in a dynamic

speaker may experience appreciable volume forces, but because

the magnet is relatively rigid, the Eqs . 36-43 describe such

speakers. However, the operation of a hypothetical transducer

containing dipole or static current elements which move through

strongly inhomogeneous em fields would not, in general, be

described by Eq. 42.

Application of Maxwell's stress tensor reveals that

forces on surface charges and currents can be evaluated by

substituting the appropriate surface delta functions into

4 30
o t •H •H u U5 •H 4-> V) ^ •H

(J

X cr

1) d) iHO CD

d

•H

O

4::! <U u

m <D •H <u 0

(J x> nJ

Jh 0

13 c: to

0) to

(DO

U

H

•H

4-> c u 4->

CD

(U6 I

4-> to

to

u

nS 0

M-i •H

>-i 0)

^3 4-) to

<D

u 0 U

•H 0

nJ to

m

4-> to 0 0 0

Cj

niJ

C

C

•H 0 Im

nS 0 a

Q. 0 0 0 m

CD 0

(A

i->

d 0 m 0

4-> •H

-P

to nJ t3 =s C CD iH 0

4-> > c0 (D c

•H •H +J

(NI 0 •H13

(O cr t/) w •1—> o

•H

(->

•H

+-> to •H

'xi

<DbO

fi

03UU

•Ha

<D

O

Qg

fH

Cj

CD

5h

<D

too

<-> u

:scoucoc

•H

tn

(0 x:

to •H>

CDtoO o

CD

CD

<D

trt > o

rH 4J r <D

c:bO6

<D x:

4->

^-> CD

to • ?-i :3 x: 6 4-)

CD to

<->

1—1

to CD

CD cr

U

•H

w J3

M-I t3

CD CD

u J-i • H

$H 0

4->

CD Ti

1—1

CD

CD CD (/) :i •H

•H ;h CD 4^ tj Xi

rrt Cy t/J 4_> CD to 03

CD rH to

•(-> 1 CD c»

10

CD •H

CD to

c! cr

to 0 w

f-4 x; 0 Ph CD 0 > u 5h •H U

•H :3

CI 6 4-> 0 •H u Jh +-> CD (/) c to •H C 0

CD 0

Ti

c:

C CD 0 H 0 n3

5-1 ft

CD 03 N TJ

03O

•H

to

CD IW

CD •HmuHu

CD

Xl

4->

toO

+-> u

CD>

CD

<Dft

+->

tooHPUG

13

<4-l O

4->

toTic

03

03 to

•H

CDX

OS iH

O,

</) •H

>^

r-t

+-> c!

CD •H

CD>dou

CD

ft

03

todo

•H

03cr

CD

CD

CD

ft rd

+->

03d

•Hm

CD rd

00 o 1:3

CD •H

<+-l

*-> •HOO

i-(CD>

CD

4-> d

03

IE

to

03e

fto

M-tX

•H

ft M

036o

03^

CDO

to 3

•H

to

O CD Q.

IpqX

lit) V / +

(>

IIP3

•H+X>

II

D3H 3

•H

Iw ta>

ll<^ 3

•H

IIP3

•HI XO

l»-3 X

to IMH o S'

Q.3

•HI X

l«° O CD Q. +

ImX

lit) X

Icq

Ilea >O CD CI

lit) X c Im+

IIS

bO • 00 ft •H 0 0

to 4_> mh 0.r—M

•H • nS to oS cj (t

*—1

o3

W •H rj > rH > ct3 Mh •H :3 rH lu •H 0 CD rH to

CD t/1 TJ CD •H :3

C5 CD t/i >^ CD to

o3 P •H ,f~i o3 CO 4->

4_> en o3 CD oS • H p tj > > CD ft C5 CD 1 a CD OS ft

CD 0 0

(J CD CD •rH •H ^ Ti 4_) P o3 CD o3

4_> tJ 0

CD to ;3 Mh cr to ^

fH o3 t3 CD {t 0 0 n3 0

4^ •H

u CD nd •rH

t/) (J 4_>

X >

(t

>—I

4_> CD o3 •H CD rH 0 0$

CD > ^J ;3 0 4J (/) (D cr

4_> <D ft OS ft t/) CD oS 0 X u n CD Ti p •H r-; •H 4_) fH C30 4-> •H

4_) ^3 oS (t

r—

H

bO 03 iH

u ft P

to

d S 0 • 0 poOPO

Ti

top

•HU

•H

ft P

CD

d

•H

OS pdou

ft p

OSs

toX

to

CD P

tfl o

•HPd

OS

C/l •H

13X

CDUd

•H

CO

to rHO

XIs

in

ftop

OSft

CDP.O

OS rH

03O

to3

Mhp

MH

CDOPP

x:

bO

•H

^^eo

ft

<4Hdo

•H

Cfl «/) 0Uu

to

pde

CD rH

CDI C3^

03do

bOd

•Hp

OSft

PUoX

•H

ft p

OS6

CJI X

X

•H

ft p

OSsoupd

CD

CD rH

CD

OS a>

CDeoo

CD X

ft

CD Xpod

OS

rn 0

•HP to •H OS d0

c/) cr •H d CD P 0 03 •H p P cr

03 ft 0

•H cr Mh

CD ft

CD •H

ft

X ,d 0 P p P

CD CD

d > Pl OS

CU s

CD OS

d

U

X 0

rC CD

u P 0

ft to

CD 0

x; 0 p th

IK 0

rH

p

rH ft

d

OS •H P Iw 0 < ft P

CD ,d P p •H

•H P

CD to

d

OS

•p

(/3 ft •H

MH0XPopd

•H0PP

•HP

trt X

USto3

t/) 0

ft0Xp

t3d

OS

u0

OSd.pd00

ftp.0Xpmo0o

p.

ft

Ph0

x;p

fto

Mh0

bO

OS Pd

OS>

OSod

fto0

rHPP

to0U

13 Tio

ftP. Xu

03O

ftP.

31
The particular arrangement of elements in the matrix of

Eq. 48 was chosen because it exhibits Maxwell's equations as

a 2x2 submatrix (the upper left corner) , and because the

associated adjoint operator matrix also represents the equations of an electroacous tic transducer (the adjoint transducer).

B . The Adjoint Operator Matrix and an

Electroacous tic Reciprocity Lemma

The development of the desired generalized reciprocity

relations requires the intermediate derivation of a bilinear

divergence expression which combines, essentially, the Lorentz

24 lemma of electromagnetic theory with the acoustical reciprocity theorem (Eq. 31). The theory of adjoint operators

assures the existence of such a bilinear expression (called

the "bilinear identity"),, and provides a straightforward

method for obtaining the identity from the operator matrix of

2 5 Eq. 48. Specifically, if L denotes the linear operator

matrix of Eq. 48, the bilinear identity may be written in

differential form as

^^^2 - (l)2L^(j)^ = V-F, (49)

where denotes the "transposed" or adjoint operator associated

with L; (J)-^^ and are any two continuously di f ferentiable

column matrices containing three vectors (the tilde transposes the column matrix to the corresponding row matrix)

;

32
and F is a bilinear function of (^^,(^^ and their derivatives.

The explicit evaluation of V»F requires the determination of

a 2 5 the adjoint operator L , which is actually defined by Eq. 49.

(In Appendix II, Eq. 49 is used as a starting point to write the

electroacous t ic fields inside the transducer in terms of the

fields on the surface of the transducer.)

For a given linear operator L, the adjoint operator L

exists uniquely (a proof of the uniqueness of L is given in

Appendix I) and may be found by transposing the matrix elements

of L, and replacing the differential operators by their adjoints

Fortunately, the adjoints of the four different linear differential operators of L,

Vx, a:V, V'a'', and V'c:V, (50a,b,c,d)

are readily determined as

Vx, -V-a^«, -a^:V, and V»c^:V, (51a,b,c,d)

respectively, with the help of the associated identities,

A^' (VXA2) -A2' (VxA^) = V«(A2xA-^) (52a)

^1 V' (a»A2) I + A2 • a^:VA^ = V'(A^»a'A2) (52b) - = = ^ ^^^^^

V' (c:VA2)| -A2- [^V' (c^:VA^) = V- (A^ 'C : VA2 -A2 'C^ : VA^) .

The subscript t denotes the transposed matrix, and A^ , A2 are

arbitrary, continuously different iable vectors. In tensor

notation the transpose of ct^jj^ and '^j^j-^^y for example, used

in Eqs. 51, 52 and below is otj-j^^ and ^-^^ly Note that the

curl operator is self -adj oint , that the direct and converse

piezoelectric and piezomagnet ic operators are adjoints of

33
mo

•(-) t-* •H 5h o O

•r—>

0)

03

p 1 m to I—I -

CD

t/1 oN

(/I

U

•H c

03

!-i O

4-' r- <A o

5h(U

m

P.O

rH

U

•H 03 •P a

03 o p u (/) 03 0)

1—1(U

!h

CD P. 5h

03

CD 10 M CD •HP

03 S-H CD

4->

O

n

u

p,

03

p

d! - 4-> •H

|ca

O 1 r—1

II Td

TO CQ 1 m

I—1

(D la t/)

II (D p (/I IS CD H

•H

fH

0) Illu P SI o

lllu^

u

03

(D •H

•P

OS P

Ln

n3!C

03

CD PC •P •H o O

•1

—I u

03 03 1 CD m

1—1Mh

CD

O

CO

cn •p •H c; •HO o •1—1 p 13

05 03 M

CD CD P. > O •H

•P

I—103

u b£i CD U t:^

CD CD

5h P 03 P (/) 03 5hO P P

03

C/l fn

CD

O PhO

c/^ CD

CD

UC

CD

CD

CD CD>

•HId

T3 o c; p 03 o3 p

CDP< CD O H

TS P 03 d !-t •H bO O

•1

—>

CD t3

OS •P

CD

4-> CD rHP.ou

M-lo

CO •Pc

CD

CD pH

CD

03 •HP

CD?H

CD m

•HTS

CUo

CD Pmoco

•HP

•H

C/1 O

P.C

03 PU

03

:sOmp

bfi •H

03U

•P

c/i

CD

to

X3

X

II to" - p>

Ilia

03 • P

!h II ;> O 3

•P •H

03 1 U X

CD >

PhOP• HO

•1

—> • P

03 II

D1 CD • P P II CJ3

<4-J •H O

£=1 , O

•H 1 PU

5hP

ty) c;ou

CD A

-P • p

llcn

Q.3

•HX o O CD Cl

jpqX

II b"" X o

llu""

p3

p)

Ilea

II

H3

•H+X

nS

u

c;CDcf

CDUm

CD>

•H

00

03 P

03

4-> 03 P

c/i I—I

03

CD>

CD P

•H

03 >1 uo

•p

0$

O CD Q. CD 1 P, • O P

lib P X c o

Icq o

•1

—> • nd p OS Ills• CD J pmodo

CO •H

03P.o pd

•Ho

•1—>

OJ

CD rdP

5h

CDU

=3 Id

c/1 d

03

fH pdou

CD

C/)

03 mo

todo

•HP

OScr

CD

CD rd

+->

to

CD J3

•H

fHU

CD

fHOP

03^

CDP.OPd

nd

03

(D

+-) o S

Q. nddO (D , o

Ipq

-p

llllu

7;,lllcn to 1 fH

CD 4J

CD Ills 6

03

fH

p

OS llica P, 1

X!

CD

to Ills O

P.

C/l P d IIP

03 1 UP P

CD IIP1 •P

4->

U 11^

•H

,d P

II

d

•H

cvi

fH

CDU

Td

C/) d

03

fH P

C/5 fH

CD •P

CDe

03

fH03Ph

03d

•H

•H

fHO

CD P

CDU

OS I—I

P.

CDfH o e

nddO CD Q.

Illu"

llica

lllai

lis

II

H

IIP

11^

II cj"

lib

34
respectively. The adjoint transducer refers to a mathematically conceived transducer with hypothetical material

parameters which may or may not be physically realizable.

Taking the adjoint o£ the adjoint operator returns the adjoint operator and transducer to the original operator and

transducer. When the transposed parameters 54a equal the

original parameters 54b, the operators L and L become identical to form a "self -adjoint" operator for which the adjoint

transducer and original transducer are one and the same. In

that case B is zero (B =-B ), which corresponds to the

o o o electric -type coupling of Primakoff and Foldy.^

The expression 53 for the adjoint operator allows the

explicit evaluation of V-F. If (^^ restricted to solutions

of Eq. 48, and (^-^ to solutions of the adjoint equations

corresponding to Eq. 48 with L replacing L, i.e.

where

and

*2

"

E

u

- —a

u

Q

=

T -icoP

s s s s

ic^y^Mg ia3yX

TSOS -B xJ SOSxj^

(55a, b)

(56a, b)

(57a, b)

35
then V*F becomes, after rearrangement and use of Eqs. 41 and 43,

When the harmonic sources are set equal to zero, application

of the divergence theorem to Eq. 58 in the volume bounded

by the surface of the transducer yields a generalized

reciprocity "lemma" between the independent elect roacous tic

fields of a given transducer and those of its mathematical

adj oint

:

The name "generalized elect roacous tic reciprocity lemma"

seems appropriate for Eq. 59 since it represents a merger

of the Lorentz lemma^^ of em theory and the acoustical reciprocity theorem (Eq. 31) as applied to adjoint systems. For

self-adjoint transducers, i.e. transducers for which the transposed material parameters of Eq. 54a equal those of Eq . 54b,

Eq. 59 reduces to the corresponding Primakoff - Foldy expression^ in which a second set of fields on the same transducer

2 0 replace the adjoint fields. Auld uses this latter expression with the complex conjugate fields to derive the orthogon2 8 ality relation between different modes of piezoelectric

V-F = V- (ExH^-E^xH) - V- (u^-T-u-T^)

= E^-(J^-iwP ) + iooy H^»M +u^'{I -B xj

^ ) s s^ ^o s ^ s o s-'

- E-(Tf-ia)P^) - icoy H-M^ - u- (r^+B xj^) . ^ S S^ OS ^ s o s-^

(58)

(59)

36
waveguides and resonators. Equation 59 demonstrates that the

waveguides and resonators need not be restricted to piezoelectric material in order for the modes to satisfy orthogonality.

A more general expression similar to Eq. 59 may be

derived from the equations represented by the operator L.

Each row of the operator L in Eq. 48 could be multiplied by

an arbitrary constant without affecting the essential content

of the equations. Specifically, if the first, second, and

third row of L are multiplied by arbitrary complex constants,

c^ , C2 and c^ respectively, Eq. 59 is replaced by

-TT..rra. ^rra,,rrA n /v j . r rmSL — ^ —a- / [c. (ExH'*) -c^ (E^'xH) ] -fida + c / (T^'-u-T-u'^) -nda = 0.

Although Eq. 60 has mathematical validity and interest.

the constants c-^, and c^ must be restricted if Eq. 60

is to be used in the derivation of the generalized reciprocity

relations between a given transducer and its adjoint transducer. First, because the right side of Eq. 60 is zero,

the constant c^ , can be set equal to unity without loss of

generality. Second, must equal c-j^ = 1 to accomplish the

derivation of the scattering-matrix reciprocity relations

(Eqs . 76 below). And third, if the adjoint operator

underlying Eq. 60 is required to describe the fields of a second transducer (the adjoint transducer) , comparison of

Eq. 53 with 48 (when the constants c-j^ , and c^ are

37
included) shows that for c-j^ = = 1, the only allowable

values of are ± 1 unless both B and p° are zero. If both

3 o e and p° vanish throughout the transducers, the adjoint

operator represents an adjoint transducer for all values of

29

^3

Equation 59 already represents the case of c^ = +1

,

where the relationship between the adjoint transducer parameters and those of the original transducer is given by Eqs

.

54 .

For = -1, the operator L in Eq. 48 changes to L , the superscript (-) indicating that the third row of L has

been multiplied by a minus sign. The operator adjoint to L

describes an adjoint transducer in which the transposed

parameters

=t ^. = o ^ o a^,£^,U^,-T^,-v^,-a^,6^,-a^,3^,c^,B^,-p^ and p^, (61)

respectively, replace the original parameters listed in

Eq. 54b. The transducer of Eq, 54a (c^ = 1) converts to the

transducer of Eq . 61 (c^ = -1), and vice versa, by reversing

the signs of the off-diagonal elements in the third row and

column of L^.

As mentioned above for L, when the transposed parameters

61 assume the values of the original parameters 54b, the

operator L and its adjoint L become identical to form a self-adjoint operator for which the adjoint and original

transducer are one and the same. In that case, p° must be ' e

38
zero (p^ = -Pg)> which corresponds to the magnet ic - type

coupling of Primakoff and Foldy.^

The equations represented by the operators L and L describe the same transducer. However, the equations represented

by the adjoints L and L refer, in general, to different adjoint transducers. (Physical sources associated with the given

transducers show up as inhomogeneous terms in the operator equations. However, it should be noted that the same physical sources

show up as different inhomogeneous terms in the L and L equations.) Section III.C. shows that a transducer self-adjoint

with respect to L satisfies reciprocity, while a transducer selfadjoint with respect to L' satisfies ant ireciprocity

.

The derivation of the desired reciprocity relations can

now be accomplished with the help of the reciprocity lemma 60.

It will also be demonstrated that the SIM relations 3 and PWSM

equations 1 result as a by-product of the reciprocity derivation.

Such a demonstration is included simply to prove that the two

matrix descriptions of a transducer, which satisfies the linear

equations 36-43, need not be postulated as a definition of linearity, but follow from the linearity of equations 36-43.

The reciprocity lemma 60 for adjoint transducers may

be written as

C. Generalized Reciprocity Relations

/ [(E^xH) -(ExH^)] .rtda

A

da = 0

,

(62)

o o

39
where the plus sign refers to the operator L and the minus

sign to L . The surface of the transducer has been taken

in the surrounding fluid, so

-n'T»n = p

-A-T^-fi = p^,

except on the feed area where the contribution to the

integral is assumed negligible.

The integral over of the em fields expands to

^o^o " ^o^o / [(E^xIT)-(ExH^)].flda (63)

A ~ S

o o

(ft points into the transducer)

,

after substitutions of the modal fields normalized to unity

by Eq. 11. The contribution from the integral over -A-^-S^

vanishes if either of the following conditions is satisfied:

1) The surface A^-S^ of the transducer (but not necessarily

the surface A^-S^ of the power supply) is electromagnetically shielded so that E, E or H, H are zero on A -S . ^ ' 0 0

(Prescribed boundary conditions on the adjoint transducer

are chosen to follow those of the original transducer.)

2) The surface A^-S^ of the power supply (but not necessarily

the surface A -S of the transducer) is electromagnet ical

° ° 30

ly shielded and no em sources exist external to

the transducer -plus - source (or detector) A. Under these

conditions, the surface A^-S^ may be enlarged to a surface of infinite radius where the Sommerfeld radiation

condition or the exponential decay in electromagnet ically

lossy media demands that the integral be zero.

40
If the transducer operates at quasi-static frequencies

so that the em fields outside the transducer obey

E = -7^p, E

^ = -Vij;^ (64a)

VxH = J, VxH^ = (64b)

modal theory may not apply, but substitution of Eqs. 64 into

the em integral of Eq. 62 shows that the contribution over

A -S still vanishes. That is, no shielding is required at

0 0

e. n quasi-static frequencies of operation for the expression

V^I - V

0 0 0 0

to represent the em term of Eq. 62.

With the em integral over A^-S^ zero, Eq. 62 becomes

V^I - V ± oo oo^^^n^n'^ / (p^u - pu^) da = 0. (65) ^ ^ o Because the fields of the transducer and its adjoint exist

independently and the velocity -current variables of each

transducer may assume arbitrary values, Eq. 65 transforms

into the SIM relations by first choosing

which gives

then choosing

loi = 1. "niCr) = 0, C66a)

^0 = Kl'o *- I

da; (67a)

o

1^2 = 0' ^z*^^^ = «Cr-r'), (66b)

41
which gives

p(r) = ± V^2^r)I^ + / p^Cr,r^)u^(r^) da. (67b)

A

o Comparison of Eqs . 67 with Eqs . 3 shows that the

various spatial impedances equate as

(68)

\ = V^^, h'(r) = ± p^(r).

The PWSM equations 1 are disclosed in a similar

31

manner once the acoustical reciprocity theorem 31, Eqs. 7,

and Eqs. 4 and 8 convert Eq. 65 to

n^Cb^a^-b^a^) ± / n (K) [a (K) b^( -K) - a^( -K)b (K) ] dK = 0.

K

Since the values o£ a^ , a and a^ , a may be designated

(69)

arbitrarily and independently o£ each other, one may choose

a^^ = 1, a^(K) = 0, (70a)

and

a ^ a^TT a ^2 = 0, a2(K) = 6(K + L) , (70b)

to extract the PWSM equations from Eq. 69:

^o ^ ^ol^o ^ "^^^^ b^(-L)a(L) dL (71a)

^o a i r a bfK) = * ,(K) ^2f'^^^ " n(K) n(L)b2(K,-L)aCL) dL.

^^^^^

42
Equations 71 coincide with Eqs . 1 under the replacements,

Soo = b^i' s^i^L) = ^ n(L)b,^(-L)/n,,

(72)

^lo*^^^ " ^o^o2<^^^/^^^^ ' ^11 = n(L)b^(K,-L)/n(K) .

If the conditions o£ Eqs. 66 and 70 are applied to the

given transducer rather than its adjoint transducers, Eqs. 65

and 69 produce Eqs. 67 and 71 with the given transducer and

adjoint transducers interchanged. That is, the SIM relations

and the PWSM equations describe the adjoint transducers as

well as the original transducer.

The generalized reciprocity relations between a given

transducer and its adjoint transducers may also be extracted

from Eqs. 65 and 69 by eliminating the pressures and voltages

of Eqs. 65 with the SIM relations, and the outgoing amplitudes

of Eqs. 69 with the PWSM equations. Such a procedure yields

^o^^b-^b^^o ^o! [h^' (r);h(F)]u^(F) da - I^/ [h'(?)^h^CF)]u^(F) da

^o ^o

± / / u^(r)[Z^(r ,r)-Z (r,r )]u (r ) da da = 0 n^ o^ o' * 0-'

^ n^ o-' o

o o (73)

43
t3

03

O

1^

£03

03

1^

1^

+

c

/—

\

oo

ino

03 O

CO

03 O

03O

cr

03

O

II

/—

\

V /

I

I

I CO

I—

I

•5t

03

+->

CD P

03 u 03 43 u T3

LO LO LO LO

!>-

03 O

03

1^

+1

cr w

o

m

o

o

+->

t/)

Du

c! •H

CO

OS

1 1 O CO 1—1 1—

1

1 m £

o 1—1 o o o £ CO CO

o3 t—

1

M

o

CO CO m o S3

/—

\

o o

:3 ISI +1 +1 ISI CO +1 +1 fr

1—1

1—1 03>

II II II II II II II II

1 + 1 + o3 1—

1

03 X o £

/—

\

/—

\

CO to £ £

03 O

5-1 CO £ £ Q 03 CCS o rH o

V ' ?H 03 XI 03 O o3 1—1 1 •P CO CO

03 O O •H 03 O O I—

1

CO CO X) ISJ S3 03 1—1 O o £

!h CO c

03

03

03

Si

cr

44
With the substitution of pressure and normal velocity

from Eqs . 7 and the help of the spectral impedance equations

24b and 26, Eq. 65 shows that the spectral impedance matrices

also satisfy

^oo = ^00 C77a)

^ol^^^ = - ^lo^-^^ C77b)

^lo^^) = * ^01^-^^ (77c)

Z^^(-K,L) = Z^^(-L,K). (77d)

The procedure and conclusions of Section II. B demonstrates

the equivalence of the three separate expressions of reciprocity (Eqs. 75, 76, 77). Either the "b" or "c" equation of Eqs.

75-77 may be considered redundant since the entire derivation

could be repeated with the given and adjoint transducers

interchanged.

When a transducer is self-adjoint with respect to L or

L , Eqs. 75-77 reduce to the original reciprocity relations

of Eqs. 2, 5, and 30. Reciprocity defined by both Kerns and

Foldy-Primakoff is satisfied by an electroacous tic transducer

that exhibits electric - type coupling (e.g. the condenser

microphone) to form a self-adjoint operator L. Ant ireciproc

ity is satisfied by a transducer that exhibits magnet ic- type

coupling (e.g. the dynamic speaker) to form a self -adjoint

operator L . If the static fields as well as the piezoelectric

and piezomagnet ic coupling coefficients vanish, comparison of

45
Eqs. 54a with 61 reveals that both adjoint transducers become

identical to the original transducer, and by Eqs. 75-77, the

transmitting and receiving parameters h, h', S-j^^, S^-j^ , Z^^,

Z^^ also vanish. In other words, there is no elect roacoustic

coupling - a result that the constitutive relations 38-42

confirm.

D . Adjoint Reciprocity Theorems

The generalized reciprocity relations 75-77 may find

expression in a variety of derivative "reciprocity theorems

and principles." As a consequence of Eqs. 75, 76, or 77,

5 32 the "elect roacoustic reciprocity theorem" ' for reciprocal

transducers extends to mutually adjoint transducers by the

relating of the microphone response of the given transducer

to the speaker response of its adjoint transducers (and vice

3 3 versa). Similarly, the "principle of reciprocity" that

relates scattered pressure at a point B from a point source

at A to scattered pressure at point A when the point source

is moved to B may be extended to mutually adjoint tranducers.

The following derivation of the aforementioned results are

performed simply and in great generality by using the Kerns

PWSM description.

The standard speaker response S(r) of a given transducer

is defined as the ratio of the pressure at a point r in the

46
ambient fluid to the transducer input current Iq>^

S(F) E (78)

I o Because incoming acoustic waves are assumed zero, i.e.

a(K) = 0, the pressure in Eq. 7a may be expressed in terms

of b(K) = VCK) alone,

p(F) = -i / b(K)e=^^^e^^-^ dK,

2/ K

(79)

The factor e"""^^ appearing in the integral of Eq. 79 allows the

z-coordinate of r to assume values other than zero. (The fluid

must contain no sources or inhomogenit ies between z = 0 and the

plane of integration.) The PWSM equations 1, which reduce to

b = S a (8Qa.^ O 00 O

^,ouc^J

b(K) = S^^(K)a^ (80b)

when a(K) = 0, combine with Eq. 4b to yield the following

expression for b(K) in terms of I^:

S, (K) I b(K) = ^

(81)

^l-^oo) %

After substitution of b(K) from Eq. 81 into Eq. 79, the

speaker response 78 takes the final form of

S(F) =

/ Sio(K)e^^*'' dK, (82)

2^(l-Soo) % K

Ic = K + Ye . z

47
The adjoint microphone response is defined as the ratio

of the open-circuit voltage for an adjoint transducer in

the presence of a spherical pressure wave centered at point

r in the fluid to the pressure in the spherical wave at a reference point r^,^

V

M^(r) = (83)

Pr^^r) = - • (84)

r - r

c (The amplitude of the spherical wave is denoted by A.) Since

I is zero, a equals b , and V is equal to 2a . The scatter

o ' o ^ o o o ing-matrix equation la for an adjoint transducer relates

a^ to a(I() ,

^o = TTT^ ! Soi(K)a(K) dK. (85)

The incoming amplitudes a(K) are given by Eq. 9a as

_ _ , _ -iiC-R

aCK) = V(K) = / p^(R^)e dR^. (86)

Replacement of the pressure in Eq. 86 by its value given in

Eq. 84 enables the explicit determination of a(K)

,

a(K) = Ai eiTZ^-iK.R^

^g^^

Y

which combines with Eqs . 85, 84, and 83 to produce the final

form of the adjoint microphone response.

48
,,a,—. 2id f ^ol ^ IFt ,jrr , ,— —

,

M (r) =

J e dK , d =

I r -r I .

Since the adjoint reciprocity relations require that

= S

oo oo

^O

Eqs . 82 and 88 show that

~ M^(r) And

OOP

' '^o (89)

S(r)

or that the magnitude of the ratio of microphone response

at r of the adjoint transducers to the speaker response at

r of the original transducer is a constant independent of the

position r and of the particular transducer involved. The

constant could also be made independent of the distance d

to the reference point if the microphone response were

"normalized" to the amplitude A of the spherical wave. If

the transducer is self-adjoint (reciprocal or ant ireciprccal

)

Eq. 89 becomes identical to the standard elect roacous tic

32 reciprocity theorem.

Next, consider the derivation of the adjoint "principle

of reciprocity" which applies to the scattering of a spherical

wave from a given scatterer (or transducer terminated in a passive load). The scattered pressure PsqC^b*^^) at a point r.

49
in the fluid caused by a spherical wave centered at a point

r^ in the fluid is given, as in Eq. 79, by

2TT ^

(90)

where b(K) may be found in terms of a(K)from the scattering

equations 1 :

b(K) =

/

L ^

SioCK)S^,(L)r^

a(L) dL. (91)

The reflection coefficient Trj, is related to the terminating

load = -V^/I^ by

o o

T o

(92)

Since the source is a spherical pressure wave centered at r^,

the amplitudes a(L) take the same form as in Eq. 87, i.e.

a(L) = A- iY(L)z^

e e o Y(L)

(93)

Substitution of Eqs . 93 and 91 into 90 results in the

scattered pressure

= - / /

K L

T oo

+ S^^(K,-L)

(94)

ir»r„ il ^ dL dK.

y(L)

50
I£ the position o£ the source and observer is interchanged and the given transducer is replaced by an adjoint

transducer, the scattered pressure at r^ is found in a similar manner to be

r.a .^_a - ^- )

Ai

Si,(K)S^^(-L)r^ ,a 1-r^s^

T 00

+ S;, (K,-L)

(95)

ik.r^ i^.rg

J j e e dL dK

27T ^ |- y(L)

(The termination for the adjoint transducers is kept at Z^.)

The adjoint reciprocity relations 76 demonstrate from

Eqs . 94 and 95 that

Psc^^B'^A^ = Psc^^A'^B^' ^96)

which completes the proof of the adjoint principle of reciprocity. (Note that the bracketed quantity in the integrands of

Eqs. 94 and 95 represents an equivalent scattering matrix

with reciprocal properties identical to those of S^-j^.) That

is, the scattered pressure measured at a point r^ from a point

source at r^ is equal to the scattered pressure measured at the

point r^ when the point source is moved to r^ and an adjoint

transducer replaces the given transducer. If the transducer

is self-adjoint (reciprocal or ant ireciprocal ) Eq. 96 ex3 3

presses the usual principle of reciprocity for scatterers

.

51
E . Power Relations

For harmonic fields, the time - average em and acoustic

power flow per unit area is given by yRe [ExH*] and - jRe[T*u*]

respectively. (The asterisk denotes the complex conjugate.)

The total time-average power P flowing into the electroacoustic

transducer through its surface is, therefore, given by

P = y Re 0 [ExH* - T-u*]- i^da, (97)

^ A

o where fi denotes the unit normal into the transducer. Since

the surface A^ lies in or on the boundary of the ambient

fluid,

fi • T* u* = "Pu^* > and the power equation 97 becomes

P = y Re 0 [(ExH*)-fi + pu *] da. (98)

^ A

^

o

If the transducer is elect romagnetically shielded, the em

part of the integral 98 evaluates simply as V^I^* by expanding E and H* in terms of the waveguide mode, i.e. from Eqs.io

E;.= Ve, H* = I*h, L O O ' t 0 0*

and making use of the normalization equation 11. (Waveguide theory shows that the basis fields (e^j h^) of a propagating mode in a uniform, isotropic guide may be chosen

re al .

)

52
c! '—V O

CTi •H txO U ^—

•H W) o

5-1

m

C e

1—t O > O Oj V) r-t •H r-H o3 +-> (/^ CD (yi •H O Oj Ph

5h 5h tJO n3 •H o o (D CD tH o '—

4-) Mh

a 03 to

03

C

•H 1 5-1 6 *^ 5-1 CIh o

+-)

—

?H CD

O 5-1 •H •H 0) •H > o3 5-< +J

+-)

H (/) CD o3

5-1 /—\

4-> <f) IS r-H 0) •H 5h CD O <D ' 5h o c CD U P. 5h • • —

o N CD c (A CD 03 p. C O CD

CD x: I-H 1—1 + &C to CD +-> Ph

O

H •(-> T)

o3 ' 5h oS •H •H •H 5-1 •H H T) O > O

oo 5-1 T} •H

O M-i 03 rc; o Cti *^ 5h 5-1 >^ > ^ o3 to CD (1) Ph O 5h

C 4S o tJ tn is CD O 1—1

CD

O o >

•H

O 1—i

P. CD 1—1 + 15h t3 (/) > O ^—

^—

CD (/) CD •H TJ o3 4-> 0) o (-> CD

Si CD ' 5h ^3 </) r—1 M •H

N </) Ph 4-> •H (/) • H CD

fx; CD i-HO pH X.U

CD 5-1 :3 1 5-1 Ph 03 :3 U to O 5-1 tn 1 5-1 O O (/) 5-1 to CD CD

0) •H ^ |5h tj o 0) •H 5h ^ ^ CJ I-H CD o + 4S O o3 /—s

to 1—<

to

C o3 rH CD CD

oJ ^ + 0 o3 </) to

•K CD n3! 5h CD to '—

- / \ J3 CD O o c U 03 'H 1—1

o3 CI rH

4-) 1 5-1 ^ l5-i (/) <+4 CJ o3 CJ

?H

—

O OS •H

C T}

CD l-H |5h Xi C/^ o3 <V CD > 1—1 \.

/ (f) • H u Ph c —

O 4-> o3 +-> •H rH o O tSI 5h rH •H •H 03 u o3 tJ p. e IS >

;=s CD 5h r-H o + o (1) CJ LO - 1 5-1 I-H X to •H CD ^ s ' o3 CD u 4-> •H * J2 C (D —

CD E o3 •H o 5h 03 CD Ph Ci H d + e O +-> 5-1 (n •H

+-> o -T)

O CD • 1—1 O <-l-l 03 u INI "—,<< CD 5-1 CD CD CD r—( ^—

+-) *-> (D •H rd

0) o o 03 CJ

M -

f-H c tx ^ 03 * •H O CD c c II + •1—I

C/) • o 1 1 H a C •H 5-1 4-) CO 03 a, o cti CD t/1 4-< •H tj 5- t4-l • U M-l

CD c 4-' o CD CD 5h o O O o3 X > *-> CD •H U 4-> CD t/i •H •H +-> 0

I-H CD 4-)

C •H c

CD p. o3 •H tJ o

5-1 e 1—1 tiC M-i c O o 03 CD 0) o o u > C n3 CJ +->

53
the spacial impedances when the transducer is lossless. In

the lossless case, i.e. when the power expression 99 is

identically zero for all values of u^ and I^, various combinations of equal to zero and equal to zero or a sum of

delta functions can be chosen to prove that the spatial

impedances satisfy the following equalities:

Z, + Zi = 0 (i.e. Z, is imaginary)

^ ^ ^ (100a)

h' (r) + h*(r) = 0 (100b)

Z^(F,F^) + Zo(rQ,F) = 0. (100c)

A reciprocal or ant ireciprocal transducer also satisfies

Eqs . 5, which combine with Eqs . 100 to show that Z^ (as well

as Z-^) is imaginary, and that both h, h' are imaginary or real

depending upon whether the transducer is reciprocal or antireciprocal respectively. Equations 29 may be used to demonstrate that no simple relations similar to Eqs. 100 exist

among the spectral impedance or scattering matrices of

lossless transducers.

Substitution of the adjoint relations 75 for the

spacial impedances into Eq. 99 yields an expression

(similar to Eq. 99) which can be manipulated slightly to

prove that the power input to both adjoint transducers (one

associated with L, the other with L ) belong to the same

54
value class, i.e. have the same definite or indefinite form,

as the power input to the original transducer. However, the

power input to the adjoint transducers does not, necessarily

remain the same as the power input to the original transducer

for given excitation I and u (r) . ^ o n

^ Finally, consider the problem of expressing the power

relation 97 in terms of the internal fields and material

parameters of the elect roacous tic transducer. The divergence

theorem transforms the surface integral 97 into a volume

integral over the volume l/^ of the transducer to produce,

P =

/ [H*-(VxE) + H-(VxE*) - E-(VxH*) - E*-(VxH)

(101)

- (V-T)-u - (V-T )-u - T:Vu - T :Vu] dV,

after the divergence is expanded, and the taking of the real

part is accomplished by adding each term to its complex

conjugate. The equations 36-43 allows the integrand of

the power integral 101 to be written as an Hermitian form.

55
IK Iw

O o

cn3

•H3

•H

"fV / 3

•H

t/5

Iw

lac +

II

O

Ills

ill^3

lib+

llto

II

w

II

w3

•H

II

H

•K 4-> lllly3

•HI

Ills3

•H

CO.

X

lit « 4-» lit) +

II

Gi

x; I X

II

a, s

•H

V)o

d) f 1 1 1

r» •rH 6

•rH

4-> ** 1 ^

•H d) t/l

(/5 Ctf . 5h o 1—1 C d) • O

r-t rrt Co •M O d)

d) •rH II V5

rH

u ,-Q <-> cti o ?-l fn o I-H ctf Irr-i •M P-i Ph

Co 4-> IX rH

fn cti •-> Ph •rH c DO d) 5h D d) T3 CD M d) •rH

t-J > Co

d) •rH •rHO

IS

i-H M o o •rH Ph d)

p-( CO ' rH

d) •rH d>

M 1—1

d) +-> o

GO •rH <U

i-H Cj 1—'

d) d) 5-1

O f-* 5-1 r>

O > d) d) •rH m > 4-> d) d)

ctf rO 5h •H ' d) W) d)

d) +-> 45

i-» 13 )-> (/) •H 1—* rJ 5h 4-> M

r-* .

d)

O

iH M-H Ct3 cd . (/) o

<DO rH ?H (/) 4—'

4-> d) to

d)

N (U •rH

O d) 4-> rH o

V) •H •s V) Pi M dJ (/) TO rH 4-> o Co M rH 1—1 M o d)

rH

r 1 1 •rH to cd IS M-l •iH O d> d)

P-i o r*

M rH •H •rH d) +J

t/l 4-> d)

u

•H Co r> ^3 d) rH

d) ^-J to

d) o C5 •rH CD

4-'

u CO cd p<

CJ

M o f-t CD fn •(-> O P( 4-> rrHC

?-i d) *^ d> •rH d> •H o r—1 u bO 4-> •H d) d) •H M-l N •rH #\ -Q

M-t t3 o •rH d) 13 #\ 5-1 5h

d) o di

U

+-> Ph

N to >^ •rH (/) d)

dJ (/) 5h •H Jh •H P.

5h > d) O > d) U rQ

(/5 P.

•HA

to

c!o

•H

4->

Cti rH

d)

5h

dJ rHP.

•H

to

c: •H

ISoo

MH

d) 45

<->

Mh

to •H

Cti t/)

56
I£, in addition, the transducer is reciprocal or antireciprocal, Eqs. 54 and 61 show that the material parameters also satisfy

a = a^ , ^=e^ , ^"^t ' ^^^t ' ^"^t * °^'^-°^t' ^'^ ^ ^t * ^^^^^

where the upper and lower sign in the last two equations o£

105 refer to reciprocal and antireciprocal transducers respectively. Equations 104 and 105 together imply that a lossless

(in the sense of Eq. 104) reciprocal or antireciprocal transducer possesses real a, e, y, c, and imaginary v and T.

The piezoelectric triadics a, a' are real for reciprocal and

imaginary for antireciprocal lossless transducers. Conversely,

the p iezomagnet ic triadics 3,3' are imaginary for reciprocal

and real for antireciprocal lossless transducers.

Equations 54 and 61 may also be used to demonstrate

from Eqs. 102 and 101 that the total power inputs for the

two transducers adjoint to the original transducer belong to

the same value class as the original transducer -- a result

which agrees with the previous analysis on the spacial

impedance power relation 99.

57
APPENDIX I

Uniqueness of the Adjoint Operator

The adjoint to the electroacous tic operator L was found

in Section III.B. And, in fact, the existence of a linear,

differential adjoint to any linear differential operator can

be demonstrated in a straightforward manner (see, e.g., Sec2 5 tion 4.17 of Lanczos.

Similarly, the proof of uniqueness of the adjoint operator when the bilinear concomitant (F) is zero can be found

3 6 as a standard theorem in many functional analysis books.

However, as far as the author is aware, the extension of the

uniqueness proof to include nonzero F has not been give^ in

the literature. For the sake of completeness, and because

the proof for differential operators becomes extremely simple

using generalized functions (in particular, the delta function) the proof of uniqueness will be given here explicitly.

Consider the equation for defining the adjoint operator

L to the given operator L:

4.^ L. .(}). - (|)iL^cf)^ = V.F((I,,(t)^) , or simply ^

i^(L({)) - i(L^(t)^) = V.F((j),(|)^) , (Al)

58
where

1) L is a given linear differential operator.

2) (}),(}) , L(j), and L cf) are n -dimens ional vector functions

of position.

3) Equation Al holds for all ^ and operable upon

by L.

3 3 3

4) L (}) is assumed to exist for the allowable (j) .

5) the dif ferent iable 3 -dimens ional vector function F

is a bilinear function of the components of and

their derivatives.

The purpose of this appendix is to show that the operator

L defined by Equation Al under the given conditions is a unique, linear, differential operator.

Linearity will be proven first. A vector function

can be added to in Equation Al to give

(A2 J

V.F((j),cf)^ + c})^) = V.[F((t,,(t,^) -H F((j),(t,^)] .

The last quantity in A2 is a consequence of the bilinear

nature of F. From Eq. Al it is also true that

i^(L^) - ICL^'^I) = V.F((D,(f)^). (A3)

Subtraction of Eqs . Al and A3 from A2 yields

~^a^a a^ ^aaaa^_

59
or

(L is linear)

since the components of ^ can be chosen independently.

To prove uniqueness assume for the moment a second

Si —

operator L2 and a second function that satisfies equation

Al. Then

i[(L|-L^)(j)^] = V-CF^-F). CA5)

Application of the divergence theorem to Eq. AS in the volume

V ^ (of the transducer in the present paper) bounded by the

surface A leads to

o

/ J [L^-L^)(f)^] dV =

/ (F^-F).h da, (A6)

o Ko assuming, of course, that these integrals exist. The surface

integral in Eq. A6 depends only upon the values of (}),<f5^ and

their derivatives on A^ . Thus cj) can be replaced by (j) + a6(r-r^)

without changing the right-hand side of Eq. A6, i.e.

^

[J + a6(F-F^)] (L2-L^)(f)^ dV = J (F2-F).n da, (A7)

o ^o

where a is an arbitrary, constant, n -dimens ional vector. Subtraction of (A6) from (A7) yields

a T a^ a (L'^-h'^^^ir^) = 0. CA8)

60
Since Eq. (A8) holds for all allowable 4> , by definition

Thus the adjoint operator is unique, as well as linear. The

differential nature of the adjoint operator is revealed in

2 5 the demonstration of existence (See Section 4.17 of Lanczos ).

Although it has been assumed that are functions of

the three position variables (x, y, z) and F is a 3 -dimens ional

vector, the proof holds for an arbitrary number of dimensions.

For an M-dimens ional space the right hand side of Eq. Al would

merely be replaced by

M 8F (i>,(t>^)

a=l 9x

a

61
APPENDIX II

Direct Integration of the Field Equations

Equation 49 of the main text can be used to derive an

expression for the electroacoustic fields at an interior point

of the transducer in terms of the volume sources and the

fields on the surface of the transducer. The derivation utilizes the Green's function to the adjoint operator and proceeds in a fashion similar to that outlined by Morse and Fesh19 3 7 bach. Section 7.5. Stratton contains an analagous

derivation for purely electromagnetic fields in homogeneous,

isotropic, nonconduct ive media. A lucid discussion of the em

3 8 expressions may be found in a more recent article by Tai

.

The purely acoustical derivation for a homogeneous, isotropic,

nonviscous fluid involves simply a direct application of

Green's theorem (see, e.g.. Section II. A, Eq. 12).

For an electroacoustic transducer which obeys Eqs

.

36 - 43, the bilinear identify 49 may be written in tensor

notation as

(f^iL-.t}). - d^iL^.cj)^ = V.F((|),(j)^), (A9) .

where the operator L^j and its adjoint L^j are given in Eqs.

48 and 53 respectively, is defined by

62
fE 1 X

E

V

E

z H

X

H

yJ H

z u

X

y

^ z

^

E

H

and F is given in Eq . 58.

Sources within the transducer may be expressed as in

Eqs . 55 -57 , i.e.

L (f)

- = Q. . (AlO)

Moreover, let represent the Green's function to the adjoint

operator , i.e.

i = l ,2 • • -9

Jl=l ,2 • • '9

(All)

Substitution o£ Eqs. AlO and All into Eq. A9 and integration over the volume \1 ^ of the transducer yields the desired

expression for 6

^ Ax ) = / Q. dV - / n-F((l),G^)da (A12)

o o Equation A12 determines the rectangular components of

E, H, and u inside l/^ in terms of the volume sources j ^3 >

63
M , r ), the fields (E, H, u, T) on the surface , and the

appropriate Green's function. It can be written in more ex

plicit form as

E

H

uI

where

and

p a ^12 ^22

/

V.

91

p a r" 3-

s s

ISOS -B xj

L

dV - / n A_

'99 ^'^*''9^J (A13)

da

,

F(<i),G^) ExH^ - E^xH - u^.T + u-T^

pa _

u

fciVu^

T^a n-a

10)

+ a .EJ +

The remaining fields may be found directly from Eqs. 38-43.

Expression A13 seems quite complicated because it applies to

such a general linear medium. However, for most transducers

all but a few of the material parameters are zero and the expression simplifies greatly.

If the sources are zero and the boundary conditions on G^

are specified as tangential = 0 and T^»n = 0 on A^, then

64
the elect roacous tic fields are determined solely by E and ^ ' tan

T«n on the surface . A similar statement holds for zero

—a -T-ra CEl,tan' "-("I-Til' [Hl,ta„> or [H^_^^„, n.(u».T)]. Of

course, it is assumed that the Green's functions exist under

the specified boundary conditions.

If all electroacous tic coupling is set equal to zero,

the transducer equations decouple into Maxwell's equations

and the momentum equation for inhomogeneous , anisotropic

material. Then Eq. A13 may also be decoupled and rearranged

to give

—a

ECr ) =

/

=^a dV - 0 (nxE) .(y^^.VxG^) da (A14)

1/ A

with

H(r') = y'^ yxE

ico V • E

=-1 u

'^o^ s

and the dyadic Green's function G^ satisfying

Vx

r=-l

^•Vx^^

loj E

where

+ b^.yxG^ + Vx(d^.Gg) + e^.Gg = 6(r-r )I , ^a

nxG^ = 0 on A

E o

b = -y '"•V, d = T'y , e = itjoe " - ia)T»y "'"•v

= Jg-iojPg + iwy^T-y"-^ -Mg + y^ Vx(y'^.Mg).

65
H(r ) =

/ F„.g2 dV - ^ (nxH) . (a'^.VxG^) da

1/ " " A

z n

o o

(A15)

with

E(r') = a"-"- • (VxH+iwx) + a"""" • ( - iojP^

)

and the dyadic Green's function tl^ satisfying

6 (r-r' )

I

where

=a nxGu = 0 on A

H o

a = a^-iooe, F = iwa '''•t, H = -iwva ^

e = -ioj(y + i(jov*a '^•t)

= iooy^M^ + iwv (J^ -ia)P^) - Vx(J^-icoP^)

o s

uCr ) = / f -G^ dV + J— / (T.n) 'G^+n* ^ ' u

rU* C

10) u (A16)

with

TCr') = c:Vu

10)

and the dyadic Green's function G^ satisfying

i- V-?^:VG,^ - io)p°^f = 6(F-F')T.

ico t u m u

^

Equation A14 and A15 represent generalizations of formu37 las (19) and (20) in Stratton to inhomogeneous , anisotropic,

magneto -electric , conductive material. Note that the electromagnetic fields inside a sourceless volume is still determined

66
by the tangential electric field alone or the tangential magnetic field alone on the surface of the volume. When the

volume includes all space, and all sources lie in a region of

finite extent, the surface integrals in Eqs . A14-A16 vanish.

(The volume integration which is left in Eq. A14 is similar

7.0 to that derived recently by Kong for "bianisotropic" , i.e.,

magneto -electric , media.)

Equation A16 shows that u and T inside a sourceless

acoustic medium are determined by T»fi alone or by fi»(u»c")

=a —a =

alone on the surface A^ provided T^^'fi or u*(u^^»c) (i = l,2,3)

are set equal to zero on the surface A^ respectively.

67
ACKNOWLEDGMENT

The author is grateful to Dr. D.M. Kerns for the original

suggestion of the work and for the many helpful discussions

throughout its completion. Thanks go to Miss Janet R. Becker

for typing the hieroglyphics.

68
REFERENCES

D.M. Kerns, "Correction of Near-Field Antenna Measurements Made with an Arbitrary but Known Measuring Antenna,

Electron. Lett., 6, No. 11, pp. 346-347, May 1970.

D.M. Kerns, "New Method o£ Gain Measurement Using Two

Identical Antennas," Electron. Lett., 6_, No. 11, pp.

348-349, May 1970.

Much o£ the theory was published originally by D.M. Kerns

and E.S. Dayhoff, "Theory of Diffraction in Microwave

Interferometry , " J. Res. Nat. Bur. Stand., 64B , No. 1,

pp. 1-13, March 1960.

D. M. Kerns, "Scat tering -Mat rix Description and NearField Measurements of Electro -Acous tic Transducers,"

National Bureau of Standards ; unpublished, July 1972.

L.L. Foldy and H. Primakoff, "A General Theory of

Passive Linear Electroacous tic Transducers and the

Electroacoust ic Reciprocity Theorem. I," J. Acoust. Soc.

Amer., 17, No. 2, pp. 109-120, Oct. 1945.

H. Primakoff and L.L. Foldy, "A General Theory of Passive

Linear Electroacous tic Transducers and the Electroacoustic Reciprocity Theorem. II," J. Acoust. Soc. Amer.,

19_, No. 1, pp. 50-58 , Jan. 1947.

E. M. McMillan, "Violation of the Reciprocity Theorem in

Linear Passive Electromechanical Systems," J. Acoust. Soc

Amer., 18, No. 2, pp. 344-347, Oct. 1946. The useful

69
term "ant ireciprocity" was apparently introduced by

McMil Ian

.

[8] Foldy and Primakoff refer to Eqs . 3 as the "linearity relations." Here they are referred to as the

"spacial impedance -mat rix relations" to avoid confusion

with the spectral scattering-matrix equations 1,

which are also linear.

[9] P.F. Wacker, "Theory and Numerical Techniques for

Accurate Extrapolation of Near-Zone Antenna and Scattering Measurements," National Bureau of Standards, unpublished, April 1972.

[10] A.C. Newell and D.M. Kerns, "Determination of both

Polarisation and Power Gain of Antennas by a Generalised

3-Antenna Measurement Method," Electron. Lett., No.

3, pp. 68-70, Feb. 1971.

[11] Between A and there can be finite regions of either

acoustically rigid material or material in which volume

forces are zero and stress is linearly related to strain

by a symmetric Hooke's tensor. However, the reference

plane S^ must remain entirely in the homogeneous ambient

fluid.

[12] R. Courant and D. Hilbert , Methods of Mathematical Physics

( Interscience , New York, 1953), Vol. 1, Chap. 3, pp.

140-142.

[13] Note the difference between " spectral " and " spacial

impedance matrix."

70
[14] The acoustical reciprocity theorem derives from the

divergence theorem and Eqs . 6.

[15] Here, the delta functions are two-dimensional since

they depend only upon the two parameters needed to

specify the surface A.

[16] Physically, the assumption that F(K,r) does not have

zero as an eigenvalue is necessary to insure a unique

value of for given and I (T) . Mathematically,

the assumption gains support from the facts that F(K,r)

is nondegenerate (nonseparable) and the infinite domain of the double variable K" is equal to or greater

than the domain of the two parameters in r describing

the surface A.

[17] D.M. Kerns, "Plane-Wave Spectra and Spectral Coupling

Products for Arbitrary Radiating and Scattering Systems,"

National Bureau of Standards, unpublished, March 1972.

[18] D.M. Kerns, "Plane-wave Scattering Matrix and Generalized

Reciprocity Relations for Antenna and Scatterers,"

National Bureau of Standards, unpublished, June 1972.

[19] P.M. Morse and H. Feshbach, Methods of Theoretical

Physics (McGraw-Hill, New York, 1953), Part 1, Sect. 1.6.

[20] B.A. Auld, "Application of Microwave Concepts to the

Theory of Acoustic Fields and Waves in Solids," IEEE

Trans. Microwave Theory Tech., MTT -17 , No. 11, pp. 800-

811, Nov. 1969.

71
[21] T.H. O'Dell, The Electrodynamics of Magneto-Electric

Media , (Selected Topics in Solid State Physics, E.P.

Wohlfarth, Ed., Vol. 11), (North -Holland/ American

Elsevier, New York, 1970).

[22] D.A. Berlincourt, D.R. Curran, and H. Jaffe, "Piezoelectric and Piezomagnet ic Materials and Their Function

in Transducers," in Physical Acoustics , W.P. Mason, Ed.

(Academic Press, New York, 1964), Vol. lA, Chap. 3.

[23] R.N. Thurston, "Wave Propagation in Fluids and Normal

Solids," in Physical Acoustics , W.P. Mason, Ed. (Academic Press, New York, 1964), Vol. lA, Chap. 1, pp.

11-18.

[24] P. Lorrain and D.R. Corson, Electromagnetic Fields and

Waves (W.H. Freeman, San Francisco, 1970), 2nd ed.,

Chap. 10, pp. 446-448.

[25] C. Lanczos, Linear Differential Operators (D. Van Nostrand, London, 1961), Chap. 4.

[26] Determination of adjoint differential operators as implied by the bilinear identity 49 is not synonymous

with the interchanging of rows and columns of the Cartesian matrix form of the operator. Contrast, for

example, Lanczos. Ref. 25 with Auld, Ref. 20.

[27] The terms "adjoint operator" and "adjoint transducer"

refer to two distinct mathematical concepts. The

adjoint of a given operator can be proven to exist

72
uniquely, (see Appendix I), whereas the adjoint transducer

finds meaning (mathematically) only because the elements

of the operator L could be arranged in the special way

that allowed the equations of the adjoint operator to

coincide in form to those of a second (physically hypothetical) transducer.

[28] J. A. Lewis, "The Effect of Driving Electrode Shape on

the Electrical Properties of Piezoelectric Crystals,"

Bell Syst. Tech. J., 4£, No. 5, pp. 1259-1280, Sept. 1961.

[29] The restriction to adjoint operators that describe an

adjoint transducer is unnecessary for the development

or utility (at least in some applications) of the

generalized reciprocity relations. The scatteringmatrix description, which depends upon the equations of

the fluid outside the transducer and upon the linearity of the transducer, remains valid for an adjoint

operator regardless of whether the adjoint operator

represents a transducer. The somewhat unnecessary restriction to adjoint transducers is introduced simply

to allow the comparison of the given transducer to a second transducer rather than to an abstract linear

system alone, and also to facilitate the discussion of

self-adjoint transducers.

[30] Regions of linear em material are permitted outside A.

73
[31] Alternatively, the results o£ Section II. A could be used

to derive the PWSM equations 1 directly from the SIM

Eqs. 67.

[32] American Standard of Acoustical Terminology (American

Standards Association, New York, 1960), SI. 1-1960, Sect.

7.30, pp. 32-33.

[33] P.M. Morse and K.U. Ingard, Theoretical Acoustics (McGrawHill, New York, 1968), Chap. 8, p. 405. Adjoint reciprocity principles for acoustic scatterers derive entirely

from a scattering matrix having the symmetry properties

of S-j^^ (Eq. 76d) . The analagous symmetry or "reciprocity theorem" exists for the scattered plane wave

amplitudes of quantum mechanics (See Ref. 19, part 2,

Chap. 9 , pp. 1130-1131)

.

[34] L. Mirsky, An Introduction to Linear Algebra (Oxford,

London, 1961), Chap. 13.

[35] J. Mercer, "Functions of Positive and Negative Type,

and Their Connection with the Theory of Integral

Equations," Proc. Roy. Soc, Ser. A, 8^, No. A559 ,

pp. 69-70, Nov. 1909.

74
[36] See, e.g., M.H. Stone, Linear Transformations in Hilbert

Space and their Applications to Analysis , (American

Mathematical Society Colloquium Publications, New York,

1932) , 15 , Theorem 2.6.

[37] J. A. Stratton, Electromagnetic Theory , (McGraw-Hill,

New York, 1941), Section 8.14).

[38] C.T. Tai, "Kirchoff Theory: Scalar, Vector, or Dyadic?",

IEEE Trans. Antennas Propagat . , 2_0, No. 1, pp. 114-115,

Jan. 1972.

[39] J. A. Kong, "Theorems of Bianisotropic Media," Proc. IEEE,

60, No. 1, pp. 1036-1046, Sept. 1972.

75
FORM NBS-114A (1-71) U.S. DEPT. OF COMM. 1 . PU BLIC ATION OR RE PORT NO. 2. Gov't Accession BIBLIOGRAPHIC DATA NBSIR 73-329

SHEET

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Theory of Adjoint Reciprocity for Electroacoustic Transducers

5. Publication Date February 1974

6. Performing Organization Code

7. AUTHOR(S)

Arthur D. Yaghjian

8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS, Boulder Labs.

DEPARTMENT OF COMMERCE

Boulder, Colorado 80302

10. Pro ject/Taslc/ Work Unit No. 2721277

11. Contract/Grant No.

12. Sponsoring Organization Name and Address Same as Item 9.

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16, ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literatiue survey, mention it here.) Analytical techniques for the measurement of the external characteristics of electroacoustic transducers have been developed by D. M. Kerns using a plane -wave scatterin

-matrix (PWSM) formulation, Foldy and Primakoff, in their classic papers on linear

electroacoustic transducers, utilize a spatial impedance -matrix (SIM) formulation, Botjh

formulations involve a continuous, linear "matrix" transformation in which reciprocity

is defined as a relationship between elements of the matrix.

The first portion of the present report demonstrates that a transducer satisfying the

"SIM relations" also satisfies the "PWSM equations" (but that the converse theorem

does not hold), and that the alternate expressions of reciprocity are equivalent for

transducers that obey both formulations.

The second portion of the report examines the equations which characterize the

internal behavior of a large class of electroacoustic transducers, A linear operator

approach is employed to develop a generalized reciprocity lemma which is used to establish adjoint reciprocity relations between the fields of a given transducer and its adjoint

transducers. The linear operator approach facilitates the identification of self-adjoint

(reciprocal or antireciprocal) transducers, and the adjoint reciprocity relations have

utility in the extrapolation techniques of the PWSM formulation. An adjoint "reciprocity

theorem" and "principle of reciprocity" are derived from the generalized reciprocity

relations. Finally it is shown that the total power inputs for the adjoint transducers

belong to the same "value class" as the original transducer.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Adjoint operators; electroacoustic transducers; reciprocity; scattering matrices.

18. AVAILABILITY STATEMENT

rxjqjNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

22. Price

USCOMM-DC 66244.P7I

USCOMM - tRL