HYDRODYNAMICAL RELATIONS 53 



Solving the three sets of equations for the excess pressure and particle 

 velocity ratios in the waves gives 



(Z.o4j — — 



r PgoCgo COS a -f- poCo COS ag U 



Pg _ 2pgoCgo COS a _ PgoCgo Ug 



P PgoCgo COS a + PoCo COS ag poCo u 



It is readily verified that for a rigid boundary {pgoCgo ^ poCo) these rela- 

 tions reduce to 



P' = p. u' = u 



Pg = 2P, Ug = 



as already found. 



A question of obvious importance is what level of compression in 

 water is compatible with the assumption that the waves are of infinitesi- 

 mal amplitude so that Eqs. (2.34) can be applied, rather than more 

 complex calculations taking into account variations of density and com- 

 pressibility. The answer can only be demonstrated by such calcula- 

 tions but a safe rule based on them is that for pressures of less than 

 10,000 lb./in.2 and angles of incidence not greater than 30°, the de- 

 partures from a regular scheme of reflection as depicted in Fig. 2.8 and 

 calculations using Eqs. (2.34) are insignificant. In the majority of 

 cases, reflections from the surface or bottom of the sea can be treated in 

 the acoustic approximation. The reflection of a pressure w^ave from a 

 free or nearly free surface presents, however, some problems of its own, 

 which we consider briefly before returning to effects in waves of finite 

 amplitude. 



C. Free surface. In the discussion of boundary conditions it has 

 been tacitly assumed that the pressure waves involved were of con- 

 stant strength at the initial discontinuity, and that the pressure at all 

 points behind the front had a constant value, i.e., the pressure was a 

 step function. With these assumptions a wave of compression strik- 

 ing a free surface is reflected as a rarefaction of equal strength which 

 leaves behind it fluid at the initial pressure. In actual cases, either or 

 both of these assumptions is likely to be a poor approximation. Con- 

 sider first the common case of a spherical shock wave of relatively small 

 amplitude. In the acoustic approximation, the amplitude falls off in- 

 versely as the distance from the source and the profile behind the front 

 is roughly an exponential decay. If this w^ave strikes a free surface 

 obliquely, we have to expect from Eqs. (2.34) that at the boundary a 

 reflected wave is set up of amplitude equal to the incident wave, but of 



