shock developed at the collapse of a vapor cavity must begin with a consideration of 

 the effect of the compressibility of the liquid upon the motion. Various extensions 

 of acoustic theory in which the wave equation in the velocity potential has been 

 retained have provided relations which can be said to give "first order" corrections for 

 the effect of compressibility of the liquid [25]. These results are useful in the con- 

 sideration of the collapse of the gas globe formed by an underwater explosion, where 

 the flow velocities do not exceed one-tenth of the velocity of sound and the pressures 

 do not exceed about six percent of the compressibility modulus of the liquid. In the 

 case of a collapsing vapor cavity, however, the significant effects of compressibility 

 are manifest while the cavity contains essentially only condensing vapor: the small 

 amount of gas inside is not sufficient to arrest the collapse before the inward flow 

 velocities exceed the velocity of sound [26]. The problem can be put then, in idealized 

 form, as the calculation, first, of the motion and, then, of the surrounding pressure 

 and velocity fields, in the case of the collapse of an empty spherical cavity in a 

 compressible liquid [27]. 



The equations of compressive flow do not, in general, admit of explicit solu- 

 tions. Gilmore [28, 29] however, has derived an equation relating the radius of the 

 cavity explicitly to the velocity and acceleration of the wall of the cavity. In the special 

 case of the empty cavity, Gilmore's differential equation is 



RR(1 - R/co) + %R 2 (l - R/3co) = - Po/po, (14) 



which reduces to the corresponding form of (12) for vanishingly small values of 



R/c . For the initial conditions considered by Rayleigh in the incompressive case, 

 Gilmore gives 



RiV / R Y / S P oR 2 \ 



R I V 3c / V 2P / 



which also reduces to the corresponding incompressive solution, Eq. (13), as R/c — > 0. 

 Gilmore's results are derived from a hypothesis of Kirkwood and Bethe which states 

 that in the spherically symmetric flow about the cavity the quantity r(h + V2U 2 ), 

 which in isentropic flow is exactly equal to r<£, is propagated outward with variable 

 velocity (c + u). Here r is the radial coordinate; c is the local value of the velocity 



of sound and <£ is the time derivative of the velocity potential. The enthalpy h is 

 defined in Fig. 9. Where the pressure in the fluid differs from the ambient value by 

 only a few atmospheres, h is simply (p — P )/p , but where the pressures are not 

 negligible in comparison with the modulus of compressibility, pc 2 , of the liquid (about 

 21,000 atmos for water), the value of h depends also on the relation between the 

 pressure and the density of the liquid. (In the present discussion, c and p must be 

 considered variable; their values at ordinary pressures (near zero) will be denoted by 

 c and p ). Here, p and u are the pressure and the radial flow velocity. 



The sound pressure, p s , at some large value of the radial coordinate is, accord- 

 ing to the theory, determined as follows: The value of the propagated quantity 



r (h -4_ 1/2 u 2 ) is easily evaluated at the wall of the cavity as RR 2 /2. The "outgoing 

 path" of each of its successive values may be traced through the pressure and velocity 

 field surrounding the collapsing cavity according to the rule given by the Kirkwood- 

 Bethe hypothesis. The assumed behavior is illustrated qualitatively in the diagram, 

 Figure 9. Several outgoing paths or "characteristics" are shown, each corresponding 

 to a different value of the propagated quantity, the particular value being the value of 



RR 2 /2 at the instant at which the path "left" the wall of the cavity. The slope of 

 the line representing each such outgoing characteristic represents, at each point in 



254 



