HYDRODYNAMICAL RELATIONS 47 



More detailed investigations of the thickness of shock fronts have 

 been made by a number of writers. In a classic paper, Becker (7) ob- 

 tained exact solutions of the hydrodynamical equations for a plane 

 shock wave in an ideal gas, and came to the conclusion that the thick- 

 ness of the shock front became comparable with the mean free path 

 for only moderate pressure ratios on the two sides of the front. Under 

 these circumstances the hydrodynamical equations are no longer valid 

 and kinetic theory must be used, which reduces to hydrodynamics if 

 fluctuations over distances of a free path can be neglected. Becker 

 further concluded that even classical kinetic theory was inapplicable 

 to very intense shock waves in gases. Thomas (113) has shown that 

 this conclusion is in error because of faulty assumptions about the co- 

 efficients of viscosity and thermal conductions for gases at high tem- 

 peratures and pressure, and when the proper values are used he finds 

 that kinetic theory should be applicable even to extremely intense 

 shocks. All these results are for gases and, of course, do not apply 

 directly to liquids. However, it seems clear that the thickness of 

 shock fronts is so small as to be virtually undetectable by direct meas- 

 urement, and the gradients of pressure and particle velocity are so 

 large that hydrodynamical methods are no longer applicable to their 

 detailed description. 



2.8. Conditions at a Boundary between Two Different Media 



In our review of the hydrodynamic conditions in fluids, we have so 

 far considered only the development and propagation of pressure waves 

 in an infinite medium. Even this development is incomplete unless 

 account is taken of the way in which conditions in the gas sphere are 

 related to those in the water at their boundary, and in practical cir- 

 cumstances the medium of propagation is of finite extent. The natural 

 boundaries which always exist are of course the surface of the sea and 

 the bottom, and the most important use of underwater explosions has 

 long been for attack against ships or other structures. The general 

 problem of what happens when the disturbances set up by an under- 

 water explosion come in contact with a different medium, at a boundary 

 is, therefore, one of considerable importance, but only a few of its 

 ramifications can be dealt with here. 



The approximation is again assumed that the motion of water 

 or gaseous products is described sufficiently well without viscosity or 

 other dissipation mechanisms, except at vanishingly narrow fronts or 

 discontinuities. If we consider a plane boundary between two media 

 having different mechanical properties (density, compressibility), it is 

 evident that the pressures must be equal at adjacent points in the two 

 media on either side of the boundary. Further, any motion of the 

 fluids at right angles to the boundary must be such that the com- 



