Bow Waves and other Non-Linear Ship Wave Problems 



For a plane Interface and 

 variables indicated in Fig. 1 we 

 have 



;i y 



8v 9p 



interface 



8t 



ay- 



For negligible gas density Taylor's 

 resxilt is! 



fluid 



gas 

 Fig. 1 



stable: 



lr<°= 



ay 



instable: 



av 



ay ^• 



marginal 

 stability: 



ay 



The question is whether this criterion {Bp/dy = 0) holds also for 

 more general interfaces. As an example consider the spherically 

 symmetric implosion of an empty gas bubble (Fig. 2), The equation 

 for the radius R is 



RR + 4R = - ^ 

 2 p 



where p^j, is the pressure far away 

 in the fluid, Pa, > 0. From this 

 relation 



RR=-Pai_4R <o. 

 P 2 



fluid 



Fig. 2 



The inviscid equation for the only velocity component v is 



av 



W "^ 



av _ i ap 



p "Sr ' 



Since v(r) = R R/r^ it follows that at r = R 



2R 



i ^ = - R + 

 p ar ^ R 



623 



