18 HYDRODYNAMICAL RELATIONS 



where the total derivative refers to a point fixed in and moving with the 

 hquid, not to a point in a fixed coordinate system. This is equivalent 

 to the statement that changes of density due to applied pressure take 

 place adiabatically. Therefore, for any point in the fluid at any time 

 for which dissipative processes can be neglected, the pressure is a single 

 valued function of density alone, and the law of variation is the adi- 

 abatic law found from the equations of state appropriate to the fluid. 

 Two different elements of fluid, however, may have undergone dissi- 

 pative processes at earlier times which involve significant and different 

 changes of entropy. At later times not involving such processes, each 

 element will have a single valued relation between pressure and density, 

 the exact form being determined by the change in entropy in the earlier 

 irreversible process, but these two adiabatic laws will be different. 



For example, a steep fronted shock wave, in which very large pres- 

 sures and pressure gradients exist, may result in considerable dissipation 

 as it passes through an element, but the entropy changes in successive 

 elements need not be the same, owing to loss in intensity of the disturb- 

 ance. The passage of such a wave can therefore leave each successive 

 element in a different condition. 



A more explicit formulation of the pressure-density relation for a 

 given fluid element can easily be written from the energy equation. 

 The internal energy E can be expressed as a function of pressure and 

 density as thermodynamic variables and its differential can then be 

 written as 



dE = —-dP -\ dp = - dp 



dP dp p2 



from Eq. (2.6). Solving for dP/dp, we have 



dP ^ /P _dE\l /dE\ 



dp W dp ) [dp) 



If initial values of P and p are given, and the functional dependence of 

 E on P and p is known from equation of state and thermochemical data, 

 the equation can be solved explicitly for P{p). 



2.2. Waves of Small Amplitude 



If the disturbances created in a liciuid by external sources are suffi- 

 ciently small, the fundamental equations of section 2.1 can be consider- 

 ably simplified. It will be assumed that the density p changes insig- 

 nificantly compared to its initial value po and can therefore be treated 

 as constant in terms of the form (d/dx) (pu) ; similarly the particle veloc- 

 ity is always small and terms of the form udu/dx can be neglected. The 

 equations of motion and equation of continuity then become 



