89 103 



time propagated backward, where 



in terms of the pressure p and the density p of the fluid; p^ is the density 

 and Cq the speed of sound for the undisturbed fluid, v' is the particle veloc- 

 ity, and dp/dp is to be taken along an appropriate adiabatic. The velocities 

 of propagation of Q and S differ somewhat from c^, but that is of no present 

 Interest. Thus in the medium there exists a continuous array of values of Q 

 which are advancing toward the reflecting surface, and another array of val- 

 ues of S which are moving backward. The local values of p. and v at any point 

 are related to Q and S by the equations 



p^^iQ+S), p^c^v = ^{Q- S) [177a, b] 



As the Incident wave advances, it meets zero values of S coming 

 from the undisturbed region ahead; hence in this wave, by [177a, b], /u = p^c^v. 

 Similarly, in the reflected wave, as soon as it becomes distinct from the in- 

 cident wave, Q = and // = - Pf,<^f^v. Thus, if subscripts i and r denote values 

 in the separate incident and reflected waves, respectively, 



f^^ = Po^o'^, = |Q>. ^r = - PoS^r = \S, [178a, b] 



At the reflecting surface, v = 0; hence by [177b] 



S =Q 

 which means that the arriving values of Q are continually being converted in- 

 to equal values of S, which are then propagated backward. Consequently, at 

 corresponding points on the reflected and incident waves S,. = Q,, and, by 

 [178a, b], p^ = P-, and also, since p and p vary together, 



Pr = P, 



This Is the usual law of reflection. 



At the wall Itself, however, 



^ = |(Q + S) = Q =Q, = 2/i. [179] 



where Q, Is the arriving value of Q and p^ is the value of p at the corre- 

 sponding point In the incident wave. This equation represents the approprir 

 ate generalization of the law that holds at the wall for infinitesimal waves, 

 namely, p = 2p-. 



Now if the fluid obeyed Hooke's law, the pressure p would be 



