38 



WAVE ACOUSTICS 



square of the cosine is }y^; the time average of the 

 product of the sine and cosine is zero. Thus, 



»^2pCo/ dx 



Similarly, 



_ / a" ' 



= i~ 

 \2p 



2 \dV 



(174) 



^2pcoJ dz 



Since / = (ll + ll + I^} \ we have the following ex- 

 pression for the intensity 



The relations (174) and (175) will be found useful 

 in Chapter 3 when the equivalence of wave acoustics 

 and ray acoustics is investigated. 



2.8 



PRINCIPLE OF RECIPROCITY 



The principle of reciprocity makes a statement con- 

 cerning the interchangeability of source and receiver. 

 Very crudely, the import of the statement is that if 

 in a given situation the locations and orientations of 

 source and receiver are interchanged, the sound pres- 

 sure measured at the receiver will be the same as be- 

 fore. To be true, under the most general conditions, 

 this statement has to be qualified in detail. The fol- 

 lowing is an attempt to formulate the General 

 Reciprocity Principle. 



Assume that a source of a given directivity pattern 

 b and a receiver of a different directivity pattern b' are 

 placed in a medium with a particular distribution of 

 sound velocity c{x,y,z), enclosed by boundaries of 

 any given shape with any particular boundary con- 

 ditions; let the output of the source on its axis be 

 given by an amplitude A at one yard. The receiver 

 will then record some pressure ampUtude, correspond- 

 ing to the amplitude B on its axis. Now let the source 

 be replaced by a receiver having the same orientation 

 of its axis and having the directivity pattern V; 

 assume also that the receiver is replaced by a source 

 which has the same orientation of its axis, the direc- 

 tivity pattern b, and the output A on its axis. Then 

 the new receiver will again register a pressure 

 equivalent to that of a sound wave incident on its 

 axis with an amplitude B. 



The proof of this theorem is difficult in the general 

 case, and will not be reproduced here. Instead, we 

 shall give the exact proof for the simple case of a plane 

 source emitting plane waves into a medium that 



satisfies boundary conditions of the type (139). We 

 shall then indicate, roughly, how the proof can be 

 generalized. 



Suppose the pressure is a function of y and t only. 

 The medium may be inhomogeneous, but both the 

 density and bulk modulus are assumed to be a func- 

 tion of y only. Then, the soimd velocity will be a 

 fimction of y. The wave equation for this case there- 

 fore reduces to 



df ^'dy^ 



and its sohition, by equation (160), will be 

 p(2/,<) = i{y) cos 2irft 



where \l/{y) is obtained from 



rfV 



-I- k\y)ip = 0, B = 



4ir2/2 



4ir2 



(176) 



(177) 



(178) 



dy- ' '"'^ (?{y) \\y) 



We assume the medium is bounded by the planes 

 2/ = and y = L and satisfies boundary conditions 

 of the type (139a), (139b), or (139c). 



We assume that the plane y = a \s, a, source of 

 sound. If by 4'a{y) is meant the function defining the 

 pressure amplitudes at every point of the medium, in- 

 cluding near y = a, then ^o satisfies equation (178) 

 everywhere except near y = a. That is, it satisfies 



^ + k^{y)4'a = Aiy) 



(179) 



where A (y) is very large in the immediate neighbor- 

 hood of 2/ = o, and zero everywhere else. Then the 

 magnitude of the plane source Sa, located at y = a, 

 will be defined by 



S.= ( A(y)dy= f A{y)dy (180) 



Jo Ja-S 



where it is understood that Sa is the limit of the 

 integral as 5 approaches zero. 



In the same way, we define a plane source at 

 y = b hy assuming an amplitude function ^i,(t/) 

 which satisfies equation (178) everywhere except 

 near y = b, that is, it satisfies 



dy' 



+ k'(y)^i, = B{y) 



(181) 



where B{y) is very large in the immediate neighbor- 

 hood oi y = b, and zero everywhere else. Then the 

 magnitude of the source a,t y = b will be 



B{y)dy. (182) 



b-i 



