108 



only by the factor log 2 « 0.69 from the value " -^ which 

 we would have If the bubble were a distance h from the 

 surface in infinitely deep v/ater. Thus for this case 



T = T («o) (1 - g"^og g ) 



If, on the other hand, the bubble were betv/een two parallel 

 Infinite rigid surfaces, its kinetic energy when expanding 

 or contracting v;ould be infinite; for this case T^oo in 

 the non-compressive approximation. 



In the discussion so far, nc accoiont has been 

 talten of any displacement of the center of the bubble, 

 or of any deviation from spherical shape, due to the proxi- 

 mity of the surface. Pollov/ing the ideas used in Appen- 

 dix 4, let us set 



i> (?,t) s cf> ^ 1^^ -^ l^c}>^ (11) 



P (r,t) » P,-f- ^ P, -t I^P;^ (12) 



R (^,t) - a(t)+|R^-|- ^^R^ (13) 



where R is the distance from the origin (taken at the 



initial position of the center of the bubble) to the 



boundary of the bubble in a direction making angle O 



v/ith the z axis, iviiich we shall draw from the origin toward 



the surface and normal to the latter. The equation of motion 



is 



|i.-i(v4)". ^^ (14) 



i 



