64 HYDRODYNAMICAL RELATIONS 



tion which requires comment. The fluid behind the shock waves and 

 sufficiently far from the wall (or plane of symmetry) arrives there by 

 passing through the incident and reflected shock waves in succession 

 while that near the wall passes through only the Mach stem. These 

 two parts of the fluid must have velocity in the same direction and the 

 same pressure but their other properties, in particular their density and 

 magnitude of velocity, will not be the same, as they have different past 

 histories. As a result a discontinuity of velocity, called a "slipstream," 

 is formed, which is also a density discontinuity but not a shock wave. 



Detailed solutions of this "three-shock" problem have been carried 

 out by von Neumann (116) by simple assumptions as to the nature of 

 the discontinuities at the points of intersection. The results are not, 

 however, in agreement with experimental results for air and there is as 

 yet no very clear understanding of what complications exist, von 

 Neumann has suggested that the difficulties may be largely mathe- 

 matical ones, and has expressed the opinion that the basic hydro- 

 dynamical equations are probably not at fault. The existing experi- 

 mental data for water are much more meager and do not offer any 

 appreciable help in the theoretical questions. 



In closing this section, it may be well to point out that, although 

 fluids in»general can be expected to show flnite amplitude effects of the 

 type discussed, the details of both regular and Mach reflection depend 

 very greatly on the equation of state for the fluid. In water and other 

 liquids, the existence of an "internal pressure" and a large exponent y 

 lead to small deviations from simple acoustic behavior until pressures 

 of the order of 40,000 Ib./in.^ are attained, whereas in air marked devi- 

 ations occur for excess pressures of 15 Ib./in.^ or less. 



D, Reflection at a free surface. One naturally inquires as to whether 

 the complex results obtained for reflection from a rigid surface have 

 their counterpart in reflection at a free surface. The necessary bound- 

 ary condition is that reflected disturbances must leave the fluid at the 

 free surface with its original pressure. This condition is appropriate 

 for a fluid surface beneath the atmosphere, as the pressure of the air 

 can change only insignificantly for the possible displacements of the 

 surface. These displacements are sufficiently small in shock waves so 

 that changes in gravitational potential are likewise small in comparison 

 with the energy of compression. If it is also assumed that density 

 changes by irreversible processes can be neglected, the density also re- 

 mains unchanged at the boundary. With these assumptions, the 

 function of reflected waves must be to restore the initial conditions 

 except for velocity acquired by the fluid in the process. 



In the limiting case of normal incidence, the boundary conditions 

 can ])e satisfied by a reflected wave of rarefaction travelling back into 

 the fluid which leaves the fluid behind at the equilibrium pressure and 



