INADEQUACY OF WAVE ACOUSTICS 



39 



By nmltiplying equation (181) by ^a, and equation 

 (179) by ^6, and by subtracting the latter result from 

 the former, we get 



dy- dij- 



which may be rewritten as 



f / 



d_/ a 



dy\ "a 



K6 _ , #a 



dyV"dy dy . 



^,B - i,A. 



(183) 



Equation (183) holds if ^„ and ^6 are arbitrary 

 functions of y, and A and B are defined by equations 

 (179), (181). We can integrate (183) over the entire 

 extension of the fluid between 2/ = and y = L, and 

 get 



L^ - ^^f^y = r^^"^ - ^^)^2/. (184) 



\ dy dy /o Jo 



Since ^a and \(/b each satisfy some combination of 

 ^ = or d^z/dy = at z/ = 0, and y = L, the left- 

 hand side vanishes identically, and 



(^„S - iP,A)dy = 0. (185) 



/ 



Jo 



Equation (185) is valid for all functions of ^a and ^t 

 and satisfies equations (179), (181), and the boundary 

 conditions (139). 



Since i? = except near y = b,hy equation (181), 



xPaBdy = Mb) Bdy = Mb)St 



Jb-d 



because of equation (182). Similarly, 



I ypbAdy = \j/i{a) I Ady = \('i,(a)Sa. 



Jo Ja-S 



Therefore, equation (185) becomes 



StMb) - S^Ua) = 0. (186) 



If both sources are of equal magnitude, then So = Si,, 

 and 



Mb) = Ma). (187) 



That is, if two plane sources of equal strength are 

 emitting plane waves into a "stratified" medium, 

 where the sound velocity obeys an arbitrary law, 

 and where boundary conditions are of the form of 

 (139), then the first source (at y = a) produces at 

 y = b the same acoustic pressure which the source 

 at 2/ = 6 would produce at y = a. 



It is interesting to note that we have proved equa- 

 tion (187) without solving explicitly for the pressure 

 amplitudes. We remember that even in this one- 



dimensional problem the equation (178) with bound- 

 ary conditions usually cannot be solved for ^ in terms 

 of elementary functions if the sound velocity is an 

 arbitrary function of y. However, we found we could 

 prove equation (187) merely by assuming that ^a and 

 <pb were solutions of the wave equation with initial 

 and boundary conditions, and by following up the 

 consequences of that assumption. 



In the general case, in which we no longer assume 

 perpendicular incidence on plane parallel boundaries, 

 the proof is more complex. Instead of equation (178), 

 we have the more complicated form of (165). Equa- 

 tions (179) and (181) must be replaced by equations 

 with the same left-hand sides as equation (165), but 

 with right-hand sides which are different from zero 

 only in the immediate vicinity of a particular point, 

 (xa,ya,Za) and {xi„yb,Zb), respectively. The distribution 

 of the functions A and B about these two points de- 

 termines the directivity of the source considered. 



The integration (184) must be replaced by a volume 

 integral, or rather, by an infinite series of volume 

 integrals to account fully for the two directivity 

 patterns, the left-hand sides of which can be shown 

 to vanish. From there on, the proof runs similarly to 

 the plane case. 



The foregoing remarks have apphed only to the 

 case of propagation in a perfect fluid. It can be shown 

 that the reciprocity principle holds, with additional 

 quaUfications, for propagation in a viscous fluid also. 



2.9 INADEQUACY OF WAVE ACOUSTICS 



In this chapter, we have set up a schematic picture 

 of the transmission of sound in the ocean, and pro- 

 ceeded to derive a rigorous mathematical description 

 of our schematic picture. Unfortunately, the results 

 obtained cannot be used directly as a basis for the 

 prediction of the performance of sonar gear. The 

 schematic picture is not nearly complete enough from 

 a purely physical point of view; furthermore, even 

 the simplified schematic picture can be solved rigor- 

 ously only for simple cases; and in the cases where 

 solutions are possible, the calculations are very 

 difficult. 



The physical picture is inadequate on several 

 counts. For one thing, boundary conditions like p = 

 of dp/dy = at the boundaries are only a vague 

 description of what actually happens at the bound- 

 aries. The surface is not a perfect plane, but is usually 

 disturbed and uneven, with the result that even plane 

 waves are not refiected according to the law of reflec- 



