60 



HYDRODYNAMICAL RELATIONS 



conditions. The conditions before and after reflection are indicated 

 schematically in Fig. 2.6. 



If the medium above the boundary is a compressible fluid, the 

 boundary conditions of continuity and pressure can be satisfied only if 

 a wave is transmitted into the fluid. The pressure and particle velocity 

 left behind the front at the boundary must equal the pressure and 

 particle velocity left behind the reflected wave, which is itself travelling 

 in the compressed upward moving fluid left by the original wave. The 

 situation is shown in Fig. 2.7. It is to be understood that the condi- 



1 



u 



u=0 



tu. 



TRANSMITTED 



B BOUNDARY 



tu' 



REFLECTED 



U' 



fu#o 



Fig. 2.7 Reflected and transmitted waves for normal incidence. 



tions shown are strictly true only at the boundary, which in general 

 moves after the original impact {u' 9^ 0) and that the pressure P' and u' 

 at the boundary will decay with time even for a plane shock wave. 



The relative compressions and flow velocities in the different regions 

 of Fig. 2.7 depend not only on the relative densities and compressi- 

 bilities but also, for finite amplitude waves, on the intensity of the 

 original wave. In the case of normal incidence on a boundary, we can 

 take as the boundary conditions 



P' = Po' 



U = V, 



where P', u' refer to the fluid below OB', P'„ u' g to the fluid above OB'. 

 The equalities are true at any time after incidence and hence the total 

 time derivatives, which measure the change at the boundary, must also 

 be equal : 



