SCATTERING BY A SINGLE IDEAL AIR BUBBLE 



461 



terns of .scattering, can be neglected.''- This princi- 

 pal mode of .spherical symmetry causes the scattered 

 sound to be a spherical wave centered upon the 

 bubble, which can be described by the formula, 



p, = ^e^''/'-/'^ (4) 



where p, is the pressure of the scattered wave, r is the 

 distance from the center of the bubble to the scat- 

 tered wave, and B is the complex pressure amplitude 

 of the scattered wave at unit distance; then B/r is the 

 pressure amplitude at the distance r. The intensity 

 of the scattered wave Is at a distance r is then, 



\B\^ 



/. = 



2cpr2 



(5) 



The problem now is to calculate the intensity 7s of 

 the divergent scattered wave from the intensity /o of 

 the plane incident wave. At this point it is convenient 

 to introduce the cross section as of the bubble for 

 scattering of soimd, which is defined by 



The physical meaning of o-^ is simple. As the intensity 

 of the scattered sound is given by equation (5), the 

 total scattered energy at this distance r from the 

 bubble center is 47it-/s. Thus by combining equations 

 (3), (5), and (6), 



iwr^L = crJo. (7) 



Hence, the sound energy flowing through an area as 

 perpendicular to the incident sound beam is equal to 

 the total energy scattered by the bubble in all direc- 

 tions. The bubble itself exposes to the incident wave 

 the cross-sectional area irR^. If all energy intercepted 

 by this area would be converted into scattered sound, 

 the rate of scattering by the bubble, or the energy 

 scattered per unit time, would be ttR^Io- 



Evidently, whether the scattered energy is smaller 

 or greater than the energy geometrically intercepted 

 by the bubble will depend on the ratio o-s/irfi^. The 

 incident wave excites pulsations of the bubble, which 

 are forced vibrations of the frequency / of the incident 

 sound. They will interfere with free vibrations of the 

 bubble at resonance, the frequency /r of which will be 

 computed presently. As is well known, the forced 

 vibrations of any mechanical system become very 

 intense if / is near the frequency fr of the free vibra- 

 tions characteristic of this particular system. In this 

 case of resonance the scattered energy can become 

 considerably greater than irR^h; thus the scattering 



cross section a, may far exceed the geometric cross 

 section ttR- of the bubble. 



In order to find <r„ or 4Tr |B|Vi^h the radial 

 velocity of the bubble surface vn will be computed in 

 two different ways, following the treatment in a re- 

 port by CUDWR.* On one hand, there is a hydro- 

 dynamical boundary condition which the air in the 

 bubble has to satisfy during its vibrations, namely 

 that the instantaneous gas pressure inside the bubble 

 must be equal to the external acoustic pressure. On 

 the other hand, there is a relation between pressure 

 and volume (or pressure and radius) which the air in 

 the bubble has to satisfy during its vibration, ac- 

 cording to the principles of thermodynamics. 



If the vibrations are so rapid that there is no heat 

 exchange between the bubble and its surroundings, 

 it may be assumed that the pulsations are adiabatic 

 changes of state. For such changes, it is found from 

 the first law of thermodynamics and from the equa- 

 tion of state of an ideal gas that PV remains con- 

 stant during the pulsations, where P is the air pres- 

 sure inside the bubble, V the volume, and y the ratio 

 of the specific heats, which for air is 1.4. Denoting 

 by Pa the average hydrostatic pressure in the water, 

 by Vo and Ro the volume and radius of the bubble in 

 the state of equilibrium, the condition for adiabatic 

 pulsations with small departures dV and dP from the 

 equilibrium volume and pressure can be obtained by 

 differentiating the relation PV^ = constant: 



y_dV 

 Vadt 



(8) 



dP_ dV l_dP 



K~ ~^ Vo Podt^ 



By expressing the volume of the bubble in terms of 

 its radius R, it is found that 



^0 = ~ '"'Rq , 



dV „„ dR 



dR 

 dt 



= VR. (9) 



If fi denotes the acoustic pressure and Ai the 

 pressure amplitude inside the bubble, the forced 

 vibrations of the air in the bubble are described by 



d-pi 

 dt 



V' 



== Aii 



2-n!l 



= 2-irifAie- 



^rnft 



(10) 



This internal acoustic pressure p.- is to be identified 

 with the excess gas pressure inside the bubble, dP, 

 which appears in equation (8). Hence, by substitu- 

 tion of equations (9) and (10) into equation (8) it 

 follows that 



dR 



2irifRoAi 2„y, 



:::: — e 



SyPo 



(11) 



The next step is to compute the amplitudes At 



