197 



In this discussion no account has been taken of the effects of 

 acoustic radiation of energy. This radiation will cause each expansion to be 

 somewhat less energetic than the preceding contraction, so that the emitted 

 pressure and impulse will be somewhat less. No attempt will be made here to 

 develop a more accurate theory of this phenomenon, but the total radiation of 

 energy associated with compressibility in the water can be estimated roughly. 



RADIATION OF ENERGY 



At considerable distances from the center of the gas sphere, where 

 pv^ is negligible, the pressure as given by Equation [7a] or [7b] falls off 

 with increasing r according to the same law that holds for spherical waves. 

 This observation leads to the surmise that moderate compression of the water 

 will not greatly alter the magnitude of the pressure p at any distant point 

 but will introduce the following features as characteristics of spherical 

 waves in contrast to non-compresslve motion: 



1 . a time lag corresponding to the finite speed of propagation of 

 sound waves, and 



2. a component p/pc in the particle velocity, added to the velocity 

 as derived from non-compresslve theory. 



This surmise is confirmed for small amplitudes of oscillation by acoustic 

 theory. 



In order to form a rough estimate of the energy radiated, therefore, 

 the pressure as derived from non-compresslve theory may be combined with the 

 acoustic formula for the energy that is carried off to infinity, in spite of 

 the fact that a strict use of non-compresslve theory leads to no loss of en- 

 ergy to infinity at all. In acoustic theory, the emission of radiation re- 

 sults from the component p/pc in the particle velocity and amounts to p^/pc 

 per unit area per second. Hence, to find the total amount of radiation, it 

 is only necessary to integrate p^/pc twice, first over a large spherical sur- 

 face drawn about the gas sphere, and then with respect to the time. Further- 

 more, the pressure falls so rapidly from its maximum value that the emission 

 of energy occurs almost entirely while the pressure is in the neighborhood of 

 its maximum, or while the sphere is near its minimum radius; hence a good ap- 

 proximate value can be obtained by using an approximate value for the pressure 

 that holds near its maximum. 



The amount of energy radiated per cycle by a sphere of gas for which 

 y « '+/3 is thus found to be, in the notation already employed, 



Q = 2V2 n' BSL^ 1^1^^' [111] 



