HYDRODYNAMICAL RELATIONS 33 



In the acoustic case we have shown that c = Co and the front of the 

 wave also travels with velocity IJ = Co. The time To is therefore zero, 

 and conditions at the front are therefore determined by the initial con- 

 dition of the gas sphere 



G{R, to) = Ga{0) = ao^aiO) 



For waves of finite amplitude, however, neither c nor U is ade- 

 quately represented by Co. As will be shown, a propagation velocity 

 c -{- a, which is approximately equal to the local sound velocity c + u, 

 is a good approximation to c, as we might guess from the Riemann 

 formulations of the hydrodynamic equations. 



Detailed calculations of the next section, based on the equation of 

 state for water boundary conditions at the shock front, show that the 

 value c -\- a behind the shock front is greater than U, a result which is 

 reasonable from the discussion of the overtaking effect. The retarded 

 time To is therefore positive and increases as the shock front advances. 

 The conditions at the front are then determined by the value Ga{To) at 

 progressively later times. This value of Ga{ro) must, however, be ex- 

 pected to decrease with time as the wave is emitted, it being a measure 

 of the energy in the fluid at the boundary. The earlier parts of the 

 wave corresponding to times less than r are thus progressively lost as 

 the wave travels outward. This characteristic can be thought of as a 

 destruction of these parts of the wave as they advance into the shock 

 front. The effects of dissipation at the shock front are thus implicitly 

 included in the framework of the theory, resulting from the way in which 

 G(r, t) is propagated and the boundary conditions at the front which 

 limit its advance. 



2.5. Conditions at a Shock Front in a Fluid 



The discussion in section 2.3 led to the conclusion that waves of 

 compression in a fluid develop increasingly steep fronts as they progress, 

 until the disturbance becomes so abrupt that dissipative processes must 

 be examined if any further conclusions are to be reached as to the exact 

 form of such shock fronts. Experimentally, however, it is known that 

 fronts of this kind are so steep as to be virtually discontinuous and their 

 exact shape is ordinarily of no practical concern; indeed, experimental 

 measurements would, by present evidence, be a matter of formidable 

 difficulty. Reserving, until later, any discussion of the dissipation 

 processes which must occur, we consider what can be learned on the 

 assumption, justified by experience and such discussion, that the thick- 

 ness of the shock front is negligible, and the front can be treated as a 

 discontinuity in comparison to the changes occurring behind it. The 

 equations applying to such a discontinuity were originally developed by 



