HYDRODYNAMICAL RELATIONS 55 



phenomena after an explosion, and must also be considered in analysis 

 of the effects of explosions on yielding structures; these phenomena are 

 described in more detail in Chapter 10. 



2.10. Reflection of Finite Amplitude Waves 



A. Normal reflection from a rigid boundary. In the case of acoustic 

 pressure levels the reflection from an infinite rigid surface results in a 

 pressure at the surface which is double the incident pressure. If the 

 changes in density are not infinitesimal this conclusion will not hold, 

 and an explicit calculation based on the conservation conditions de- 

 veloped in section 2.5 must be made. The simplest case is the one of 

 normal incidence and the calculations for this case will be considered 

 first. If the incident wave travels with velocity U into a fluid at rest 

 with density po and pressure Po and leaves behind it fluid with density p, 

 pressure P and particle velocity w, we have 



(2.35) piU -u) = poU, P - Po = poUu 



When this wave strikes the boundary, the physical situation must de- 

 velop in such a Avay that the fluid at the boundary is at rest. This 

 boundary condition is satisfied if a reflected wave of compression travels 

 back from the boundary with velocity U' and a strength just sufficient 

 to bring the fluid entering the front with velocity U' — u to rest. If 

 the pressure behind this front is P' and the density p' we have for con- 

 servation of mass and energy 



(2.36) p([/' - u) = p'U\ P' - P = p'U'u 



Solving Eqs. (2.35) and (2.36) for the ratio of excess pressures behind 

 the incident and reflected waves in terms of the densities gives 



(2.37) 



In order to determine the ratio a second pressure-density relation is re- 

 quired. Strictly, this should be computed using the Hugoniot energy 

 condition and equation of state data, as the passage of each shock front 

 involves a finite dissipation and leaves the fluid behind the front on a 

 new adiabatic. For water, however, the change in entropy is small 

 and an adequate approximation is obtained if the change is neglected. 

 The same adiabatic equation of state can then be used for all three 



