54 



A. ACOUSTIC R ADI ATION B y A SPHSRIGALLY 3Y:gJBTRIGAL BUBPLE 



An exact calculation of the motion of a pulsating gas 

 bubble in a compressible fluid would be very difficult: Lamb-^° 

 has derived a partial differential equation governing the 

 variation of velocity with radius and time for the spherically 

 symmetrical case, but this equation is complicated and would 

 be very difficult to solve. Fortunately, however, the effect 

 of the finite compressibility of water on the motion of the 

 bubble produced by an explosion amounts, except in the very 

 early stages represented by Fig. 1 (a), only to a small to 

 moderate correction to the simple theory of the preceding 

 section. It is therefore possible to work out the details of 

 the motion, as affected by the finite compressibility, by an 

 iterative process of successive approximations, talcing the 

 non-compressive motion as the zeroth approximation. One way 

 of doing this is carried through in Appendix 2. This method 

 involves transforming the equations of motion of the boundary 

 of the bubble into a form In which the correction for 

 compressibility is represented by a radial integral whose 

 Integrand decreases very rapidly with increasing distance 

 from the center of the bubble; in this form but little error 

 is made if the non-compressive approximation to the integrand 

 is used. The result is a differential equation — Eq. (19) 

 or (20) of Appendix 2 — for the time variation of the radius 



16 



H. Lamb, Phil. Mag. 45, 257 (1923). 



