48 HYDRODYNAMICAL RELATIONS 



ponents of particle velocity perpendicular to the boundary are the 

 same at adjacent points. With the neglect of viscosity, there is no 

 restriction on the tangential components of particle velocity and the two 

 fluids are free to slide parallel to the boundary. This lack of restraint 

 is not strictly true in any fluid, and there must be a layer at the boun- 

 dary where viscous friction is important if there are differences of 

 tangential velocity near the boundary. For the cases we shall have to 

 consider, however, there is no reason to believe that this layer is thick 

 enough to modify the motion appreciably, and in the special case of 

 motion normal to the boundary this question does not arise. 



u'/ 



COMPRESSION ' ^ ^ 



RAREFAC 



S 



Fig. 2.5 Reflection of a plane shock front at an infinite boundary. 



Before becoming involved with mathematical consequences of the 

 boundary conditions on pressure and normal particle velocity, it is of 

 interest to examine the phj^sical situation when a shock wave arrives at 

 a boundary. We consider that at some time a plane shock front of 

 compression SO is incident on the boundary BB' at a point as shown 

 in Fig. 2.5, the wave front making an angle a with the boundary as 

 shown. To the right of SO in the undisturbed fluid the particle velocity 

 is zero, but the fluid behind the front has acquired a velocity u normal 

 to the front in the direction of advance, the magnitudes of u and shock 

 front velocity U in the fluid below OB' being related to the compression 

 by the Rankine-Hugoniot conditions and equation of state. We have 

 next to consider the state of affairs along the line OB' (actually a plane 

 normal to the plane of the figure) . Suppose, first of all, that the medium 

 above the boundary is infinitely rigid, by which is meant that no motion 

 of the boundary can take place. As the shock wave travels from left 

 to right it leaves the fluid at the boundary with a component of velocity 

 u cos a normal to the boundary. This is, however, inconsistent with 

 the assumed rigid boundary and the resistance of the boundary must 

 therefore modify the motion of the water in such a way as to annul the 

 normal flow set up by the incident wave. A stationary condition can 



