102 



APPENDIX 5 EFFECT OF PROXIMITY TO A FREE 

 OR TO A RIGID SURFACE 



Lot the center of the c^s bubble at the start of 

 the notion be a distance h beneath the free surface of the 

 water, or alternatively a distance h from a plane rigid 

 surface. The v;ater v/ill be t reated as incompressible, and 

 Gravity will bo neglected, since it is easy to show that 

 the first-order effects of the surface and of gravity are 

 additive, we shall be interested in two effects due to 

 the presence of the surface, namely, the change in period 

 and the rising or sinkinc of the bubble. Tlie co;nplete 

 theory, which is siinllar to that of Appendix. 4, is rather 

 complicated; hov/ever, the calculation of the effect on 

 the period can be made quits siriiply, &nd this v/ill be 

 ^iven first. 



The bou2idary condition which must be satisfied 

 at a free surface is that the pressuro p must be constant 

 over the surface. This means that the velocity potential 4" 

 must satisfy 



-fr- -| (^t f = (i; 



on the surfac'3» Since S is of order l/r, wher© r is the 

 distance from the center of the bubble, tbo ratio of the 

 second tor-Ti of (1} to the first ivill be of order l.-'h as 

 h. —>"''■. It is not hard to show that this ratio is quite 

 onall when h is even a few times SL-^y* ^o that it v/111 be 

 20od enourh for oi^x purposes toraplac© (1) by the condition 



