37 225 



The last two of these equations are adapted from Equations [55] and [56]. In 

 writing the pressures, hydrostatic pressure is omitted. 



In these equations pj may be regarded as given, whereas p^, P3 and 

 the phase shifts t and r' are determined by the boundary conditions. At the 

 Interface, at which x = 0, the pressure and the particle velocity must be 

 continuous. By writing dovm the two equations that express these conditions, 

 and putting In them, first, t = 0, then ut = n/Z, four equations are obtained. 

 From these equations p^, Pj, r', t can be found. It will suffice to write 

 down the following formula, obtained by eliminating all unknowns but p^ from 

 the four equations : 



The coefficient of reflection K, or fraction of the incident energy 

 that is reflected. Is equal to V^ Iv-^' since reflected and incident waves 

 travel In the same medium and their intensity is, therefore, proportional to 

 p^. Hence 



222 •> 



_ (1 - ^) + ^ (1 - v) + &^ 



in view of Equation [65]. 



SCATTERING BY A SINGLE BUBBLE 



The amplitude of pressure at a distance r in the waves scattered by 

 an isolated bubble is -p^Rjr in terms of the amplitude p^ at the surface of 

 the bubble, whose radius is R^. Where r is large, the waves are sensibly 

 plane, and the average energy transmitted by them across unit area per second, 

 if they are sinusoidal. Is 



I^(^f 171] 



In which the factor l/2 represents the effect of averaging over the square of 

 a sinusoidal function of the time; see TMB Report ^+80 (4), page 39, Equation 

 [5]. The total energy scattered to Infinity per second by the bubble is thus 



^ = ^(^'4-— ^^^P/ [72] 



2pc \ r I PC '^■2- ^ ' ' 



Now upon substituting derivatives from Equation [62] in Equation 

 [59] and combining the resulting terms into a single sinusoidal term, it is 

 found that 



