85 



waves satisfy our boundary conditions, we nay write 



r'r(r,t) - f(t -£ ) , (5) 



c 



where c is the velocity of sound in water. We then .have 



(6) 



For the larce ar.plitudes with which we are chiefly con- 

 cerned the acoustical theory on which (5) is based is of 

 course invalid. It will be shown below, however, that 

 for larce amplitudes of notion tVie integral on tho left 

 of (6) can be represented by the sane term, caa"(a) , 

 plus other terms of lesser importance. 



Let us start by takin^^ tho dlverconce of the 

 Euler equation (i.e., of equation (1) without the viscosity 

 terms) and then differentiate with respect to fcimeo The 

 result is 



(7) 



"Jow 



For simplicity it will be assumed that c is not appreciably 

 chanc®^ linder the compressions to be encountered, so that 

 it can be treated ss a constant in tho differentiation. 

 This approximation is only bad at the start of the first 

 pulse, a sta,:-e which we have already excluded from the 

 present treatr.iont because at this star;e tho pressure cannot 

 be asGU-nQQ constant throughout the ''as bubble. For constant 



