HYDRODYNAMICAL RELATIONS 51 



dP^ ^ dPl 

 dt dt 



du' _ dUg 

 dt ~ ~dt 



These are the appropriate conditions, for example, in following the 

 boundary of the gas sphere after detonation of a charge. 



2.9. Reflection of Acoustic Waves 



A. Rigid boundary. Before considering the more general problem 

 of waves of finite amplitude, it is worthwhile to examine the results in 

 the limiting case of small compressions and flow velocities. Returning 

 to the case of a rigid boundary indicated in Fig. 2.5, Ave observe that, 

 for a sufficiently weak disturbance, the fluid behind the incident front 

 OS differs infinitesimally from that ahead of it. As the front progresses 

 parallel to the boundary, points on it at fixed distances all have the 

 same velocity Co/sin a parallel to OB', where Co is the normal velocity of 

 the infinitesimal disturbance OS. The postulated front OR travels 

 with the same velocity Co and the point velocity is Co/sin a where a' is 

 the angle of the front with the boundary. In the approximations of an 

 infinitesimal disturbance these velocities become asymptotically equal: 

 Co/sin a = Co/sin a', and we have the familiar law of reflection that 

 a = a. The condition imposed by the boundary is that there be no 

 net component of particle velocity normal to it, which is expressed by 



(2.33) u cos a — u' cos a = {) 



and since a = a', necessarily u = u' . Hence the particle velocities of 

 the two waves are equal and it follows that the two fronts are of equal 

 strength. 



The pressure at the boundary to the left of is the sum of the two 

 pressures and is hence 2P, regardless of the direction of the incident 

 shock. This independence of direction, however, leads to a familiar 

 acoustic paradox. If taken literally, this result predicts a pressure 2P 

 for a = 90°. But a = 90° represents a wave parallel to the boundary 

 for which there is no change in contact with the boundary as it advances 

 and no way for the pressure to be doubled. The derivation, of course, 

 breaks down for a = 90°, as cos a = cos a' = in Eq. (2.33), but we 

 are nevertheless left with a discontinuity between the value 2P for a 

 only slightly less than 90° and the value P behind a front travelling 

 parallel to the boundary. This circumstance holds strictly only for 

 waves of infinitesimal strength and the whole analysis and conclusions 

 from it may be expected to break down for waves of finite amplitude. 

 This question is examined in section 2.10, which describes the theory 



