1^6 HYDRODYNAMICAL RELATIONS 



Herring^ has also given a particularly simple derivation which illus- 

 trates the type of argument that can be used to determine the order 

 of shock front thickness given by macroscopie viscosity considerations. 

 If the coefficient of shear viscosity is denoted by ju, the classical theory 

 of viscous fluids shows that the energy dissipation per unit volume and 

 time in a plane wave is given by 



2m 



& 



For a plane wave, however, dii/dx = — (1/p) dp/dt by the equation of 

 continuity, and writing A^ = (1/c) Ax where Ax is the thickness of the 

 front, the dissipation per unit area of the front is roughly 



2m(-T) • Ax ^ 2mc^" ^^^^' 



im--- 



po^Ax 



where Ap represents the increase in density across the shock front. In 

 order to form an estimate of Ax it is necesssary to obtain an alternative 

 estimate of the energy dissipated in the front. This can be done if we 

 assume that the Rankine-Hugoniot conditions are at least approxi- 

 mately true. With this assumption the dissipated energy can be esti- 

 mated from the difference in area between the R-H curve on the P-V 

 diagram and the adiabatic curve, as Herring does. Essentially the 

 same result can be obtained by using the dissipated enthalpy computed 

 by Kirkwood and Montroll (59, II). If the ratio of dissipated enthalpy 

 to the total is a, we have 



Energy dissipated/gram = a AH = au 



(-1) 



from the fact that AH = A{E + P/p) = u{U - u/2). The mass of 

 water traversed in unit time by unit area of the shock front is with 

 sufficient accuracy poC, and equating the two expressions gives 



For a pressure of 20 kilobars Kirkwood's calculations give Ap = 0.32 

 gm./cm.^ a = 0.05, and employing a value m = 0.01 cgs. units gives 

 A X ^ 0.6 X lO-'^ cm. 



3 Herring's calculation is given in a chapter written by him on "Explosions as a 

 Source of Sound," for inclusion in a summary technical report for Division 6 of 

 NDRC, and to be entitled "Physics of Sound in the Sea." 



