464 



ACOUSTIC THEORY OF BUBBLES 



tering coefficient for frequencies considerably greater 

 than the resonant frequency /r- Since ri is less than 

 one, ■q'^ in equation (22) may be neglected when the 

 ratio /r// is much greater than one. Consequently, the 

 scattering cross section for low-frequency sound may 

 be written approximately as 



iirR^ 



(/)* = -(^')'- « 



This equation is known as Rayleigh's law of scatter- 

 ing for long-wave radiation. It will be remembered 

 that in optics Rayleigh's law explained the blue color 

 of the sky, as the resonant frequencies characteristic 

 of the atmospheric gases oxygen and nitrogen are far 

 greater than the frequencies of visible light. Hence, 

 the shorter (blue) waves of sunlight are scattered 

 more strongly than the longer (red) waves and reach 

 our eyes with greater intensity. Equation (28) is also 

 applicable to the high-frequency sound commonly 

 used in echo ranging provided that the bubble 

 radius R is very small; if R is less than 10~' cm, how- 

 ever, fr is given by equation (26) instead of by equa- 

 tion (18). 



28.2 SCATTERING AND ABSORPTION 

 BY AN ACTUAL BUBBLE 



So far, the attenuation of sound resulting from the 

 absorption of sound energy during the pulsation of 

 the bubble has been neglected. The existence of such 

 an effect is a direct consequence of the second law of 

 thermodynamics, which implies that energy must be 

 extracted from the sound field and dissipated into the 

 surrounding water in the form of heat, in order to 

 maintain the forced pulsation of the bubble against 

 the internal friction of the bubble-water system. In 

 other words, it is thermodynamically inadmissible 

 to treat the pulsation of the bubble as if it were a 

 strictly adiabatic process; therefore it becomes neces- 

 sary to amend the analysis given in the preceding 

 section for an ideal bubble. 



This task is accomplished by adding to equation 

 (13), which expressed the continuity of pressure at 

 the bubble surface, a certain term which takes into 

 account the frictional force modifying the behavior 

 of an actual bubble. Moreover, the exchange of heat 

 between bubble and water by conduction neces- 

 sitates a modification of- equation (16), which formu- 

 lated the continuity of velocity at the bubble surface. 

 The treatment of the case of the ideal bubble im- 

 plicitly assumed that the pulsations are thermo- 

 dynamically reversible; that is, the work put into the 



bubble during compression was supposed to be equal 

 to the work done by the bubble during expansion. 

 Actually, there is heat exchange between bubble and 

 water, but the pulsations are too rapid to permit a 

 complete leveling of temperature at every instant of 

 the cycle. Thus there prevails a continual change of 

 state which is somewhere between the adiabatic and 

 isothermal case. 



It is not difficult to see that under such circum- 

 stances the pulsation of pressure cannot be in phase 

 with the pulsation of volume. While the bubble is be- 

 ing compressed, the temperature rises steadily; as 

 soon as the rise of temperature becomes appreciable, 

 heat conduction begins to operate and the bubble 

 tends to cool off even before expansion has started. 

 When the minimum volume is reached, the tempera- 

 ture will be decreasing as heat flows from the bubble 

 into the water. Consequently, the temperature maxi- 

 mum will be reached some time before the bubble has 

 been compressed to its minimum volume. Likewise, 

 since the gas pressure is proportional to the tempera- 

 ture, the maximum pressure will not be attained 

 simultaneously with the minimum volume, but some 

 time before. Thus there exists a phase shift between 

 pressure and temperature on one hand, and volume 

 and radial velocity of the bubble on the other hand. 

 For resonant bubbles at frequencies of 100 kc or less, 

 this effect is taken into account^ by inserting a com- 

 plex factor 1 — |8? in the right-hand side of equation 

 (11), where /3 is a positive constant much smaller 

 than one. 



The two equations of continuity, (13) and (16), 

 must therefore be replaced, for an actual bubble, by 

 the following ones: 



P, + Ps-Pi= -Ci— , (29) 



dt 



or 



and 



A + B/Ro - 2rt/bB - Ai = 



C,Bi 

 2TfpRl 



B 2-7rfRoAi 



(1 - fit) 



(30) 



2TfpRl 37F0 



In equation (29) Ci is a constant measuring the 

 effect of friction, which is assumed to be proportional 

 to the radial velocity dR/dt of the bubble. The term 

 CidR/dt represents the net pressure on the bubble, 

 which is positive when the bubble is contracting 

 {dR/dt < 0) ; hence, the correction term appearing on 

 the right side of equation (29) must carry a minus 

 sign. 



