SECT. 4] SOUND SCATTERING BY MARINE ORGANISMS 515 



An important special oceanic scatterer is the gas bubble in water. Theory 

 shows that the incident sound drives the gas bubble into resonant oscillation 

 at a frequency corresponding to a wavelength in water much larger than the 

 circumference of the bubble. Furthermore, near resonance the scattering cross- 

 section of the bubble is many times its physical cross-section. 



The scattering cross-section, a, is defined as in equation (1) above. Following 

 the treatment given in NDRC, Division 6 (1946), the spherically symmetrical 

 modes of oscillation of the bubble dominate, and directional modes are negli- 

 gible where the circumference of the bubble is much smaller than the wave- 

 length. Thus the scattered wave can be described by the formula 



rps = (B/r) eS-^a^-'-M). (3) 



The intensity at range r is 



Is = |5|2/2pcr2, (4) 



and for an incident plane wave can be written 



/o = |^|2/2pc. (5) 



Hence 



G = 47r|5|2/|^|2. (6) 



This ratio can be computed by applying continuity of instantaneous pressure 

 across the water-bubble boundary, and by applying the thermodynamic 

 equation of state within the gas bubble. The oscillations are assumed adiabatic ; 

 that is, pV'y = a, constant, where p is the internal pressure, V the volume and y 

 the ratio of specific heats at constant pressure and constant volume. If Pq be 

 the average hydrostatic pressure in the water, Fo and R the volume and radius 

 of the bubble at equilibrium, then small adiabatic pulsations, dV, dP, about 

 equilibrium are related by 



(7) 



where dRjdt — vr is the instantaneous velocity of the bubble boundary. 

 The acoustic pressure inside the bubble can be expressed by 



Pi = Ai e^-ift, -^ = 27TifAi e2"«/«. (9) 



The acoustic pressure is to be identified with dP, the small departure from 

 equilibrium pressure in (7). Hence substituting (8) and (9) in (7): 



dR 27:1 f RAi , .,, 



