HYDRODYNAMICAL RELATIONS 57 



theory. The differences in durations and impulse are, however, less 

 than 5 per cent for pressures up to 4,000 Ib./in.^ and so can be disre- 

 garded in many cases. It is also to be remembered that a one-dimen- 

 sional case has been considered and results of this kind hold only if the 

 boundary is of infinite extent. For a surface of finite extent, the dif- 

 ference between incident pressure at the edge of the surface and doubled 

 pressure at interior points must be relieved by a diffraction wave. In 

 other words, a solution equivalent to geometrical optics does not satisfy 

 the physical conditions, and the hydrodynamic equivalent of optical 

 diffraction must be considered in cases where the linear dimensions of 

 the surface are comparable with the length of the pressure wave. This 

 situation, which must obviously be considered in analyzing the forces 

 exerted by an explosion on a target, will be examined in Chapter 10. 



B. Oblique incidence at a rigid boundary. The discussion in part (a) 

 showed that no very considerable modifications of the simple acoustic 

 law of reflections resulted for finite amplitude waves at normal inci- 

 dence, but a further question, of what happens to the acoustic paradox 

 of a discontinuity in pressure at grazing incidence, remains to be con- 

 sidered. One might offhand expect that for finite waves the pressure 

 at the boundary would fall continuously to the incident pressure as the 

 angle of incidence approached grazing values. This, however, is com- 

 pletely untrue and the apparently paradoxical phenomena actually 

 observed have been the subject of considerable investigation, especially 

 in air where the actual conditions are particularly striking, von Neu- 

 mann (116) has considered the general problem for a rigid boundary in 

 considerable detail, and his theoretical considerations have been ex- 

 tended to specific types of fluids by Polachek and Seeger (87). Any 

 very complete summary of these investigations would be beyond the 

 scope of this work, and all that is attempted here is an outline of the 

 physical factors which lead to the complex phenomena involved and a 

 brief discussion of the results. 



Before considering in detail the incidence at a rigid boundary, it is 

 pertinent to answer an argument which might plausibly lead one to 

 conclude that the whole question is academic. This argument runs as 

 follows: the non-acoustic behavior of water is significant only for very 

 high pressures, at these pressures even materials such as steel will yield 

 and hence a rigid boundary under these conditions is an unrealistic 

 assumption. The argument is not without merit, and certainly the 

 assumption may be an oversimplification of many actual circumstances. 

 The premise that only extremely high pressures need be considered is, 

 however, not demonstrated, and in fact we shall see that it is erroneous 

 for nearly glancing incidence. A second point is the fact that reflection 

 of a shock wave from a boundary is precisely the same problem as the 

 intersection of two plane shock waves of equal intensity, the boundary 



