21^ HYDRODYNAMICAL RELATIONS 



simplest example is the perfect gas for which P = kp^ and therefore 

 c = \/kyp^~^. The ratio of specific heats 7 is always greater than 

 unity and hence c increases with increasing density. The adiabatic 

 curve for water is another example of the same law, although the den- 

 sity changes are fractionally smaller. The velocity of infinitesimal 

 waves cannot, therefore, be treated as constant, but is rather a quantity 

 which increases in regions of greater density. In order to see what 

 effect this has on wave propagation, suppose that, as a result of dis- 

 placements of a piston in a tube, a plane wave of pressure is advancing 

 from left to right in the tube, and at some instant in time has the form 

 shown in Fig. 2.1(a). Compression started in the positive direction at 

 point a will appear to travel with a speed Ca relative to the fluid at the 

 point. If the particle velocity in the fluid is Ua, the speed with respect 



(a) (b) (c) 



Fig. 2.1 Formation of a shock front in a plane wave of finite amplitude. 



to the walls will be Ca + Ua. Similarly, a compression at point 6 will 

 travel with a speed ci + Ub relative to the fixed wall. If the pressure 

 set up in the fluid by the main wave is greater at b than at a, the speed 

 of sound c and the particle velocity u will both be greater at b, and the 

 disturbance at b will advance faster than that at a. At a later time, 

 therefore, we have to expect that regions of higher pressure in the wave 

 will approach those of lower pressure ahead of it, as shown in Fig, 

 2.1(b), the effect increasing as the pressure differences increase. The 

 ultimate result of this overtaking effect will be to make the front of the 

 wave very steep as shown in Fig. 2.1(c). As the condition of inflnite 

 steepness is approached, however, the pressure and temperature of 

 closely adjacent layers will be very different; in other words, the gradi- 

 ents will be large. Under these circumstances, large amounts of energy 

 can be dissipated as heat, effects which have been neglected in the funda- 

 mental equations of section 2.1. We should, therefore, not expect 

 results based on these equations to apply to steep fronts, and will have 

 to consider this situation in another manner, as is done in section 2.5. 



Steep fronted waves of this type arc known as shock waves, and 

 while it is not physically reasonable for the ultimate slope to be in- 

 finitely steep (as this would imply infinite accelerations), shock waves 



