52 HYDRODYNAMICAL RELATIONS 



due to von Neumann for the more general case. These results are both 

 curious and interesting, and have consequences of some importance in 

 the understanding of underwater shock waves. 



B. Arbitrary boundary. In the acoustic case of an arbitrary medium 

 above the boundary we must add a transmitted wave in the medium, as 

 shown in Fig. 2.8. It is again to be expected that the angles of inci- 



Fig. 2.8 Acoustic transmission and reflection. 



dence and reflection are equal, and for a stationary condition the 

 tangential velocity of the front of the transmitted wave must equal 

 that for the incident wave, hence 



Co/sin a = c^o/sin ag 

 which gives Snell's law^ for the direction of the transmitted wave 



sm a 



Co 



sin ag Cgo 



Continuity of pressure and particle velocity across the boundary 

 requires that 



u cos a — u' cos a = Ug cos ag 



For sufficiently small disturbances the excess pressures and particle 

 velocities in the waves are related by the equations (see section 2.2) 



P = PoCoU, P' = PoCoU', Pg = PooCgoVg 



in which it is assumed that the density and propagation velocity are 

 negligibl}^ different from the values po, Co in the undisturbed fluid. 



