220 32 



This change in volume Is supplied partly by a change In the volume of the wa- 

 ter itself, partly by a change in the volume of the bubbles. As the elas- 

 ticity of water is equal to pc^, where p is its density and c the speed of 

 sound in it, the increase in volume of the water is 



—2{-^dt)dx 

 pc^ \ dt I 



where p denotes the average pressure in the water surrounding the bubbles. 

 The increase in volume of the ndx bubbles in the element is 



Q /4 „i\ ,. . „2 dR 



Hence 



■^i^ ^T7 (t "■^l rf' = 4 7rni?^ ~ dx dt 

 oM 3 / dt 



-TT- dx dt = ^ — — dx dt + AnnR'^ -, — dx dt 



ox pc^ dt dt 



dp 2 dv , 2 ,-.2 dR 



ot dx dt 



[52] 



During the same time the momentum in the layer has been changed by 



, _, dt] dx = — ■— dx dt 



^ot I dx 



since dx(- dp/dx) represents the net force on the element, whence 



dv _ _ J^ dp 



dt p dx 



[53: 



If Equation [52] is differentiated with respect to t and Equation 

 [53] with respect to i, and if d'^v/dtdx Is then eliminated between the two 

 equations, the equation of propagation for p is obtained 



Q^V 2 d'^p , 2o2 d'^R 



dt^ dx'- dt^ 



Here a term in (dR/dt)^ has been dropped as being of the second order. In 

 the same way p was treated as constant in deducing Equation [53]; and the de- 

 crease in mass due to the presence of bubbles was also ignored as being very 

 small. 



Equation [5*+] is unusual in form among wave equations in that it 

 contains two dependent variables, p and R. A second equation is, therefore, 

 necessary, and it may be obtained by analyzing the motion of the bubbles. 



This motion can be handled conveniently with the help of the prin- 

 ciple of superposition. If the bubbles did not change in radius, the inci- 

 dent wave, according to the assumptions made, would cause the pressure p near 



