68 



RAY ACOUSTICS 



where D is an arbitrary constant to be determined 

 later. Then equation (115) becomes, on dividing 

 through by D^, 







If D is chosen so that 



4t^PB 



cl 



then equation (115) becomes 



cPF 



+ (K + u)F = 



where 



du^ 



K = 



ncl 



(122) 



(123) 



{■^PBcY 



Equation (122) has solutions of the type desired only 

 for certain characteristic values of K, denoted by the 

 symbol K^. The different values of Ki are determined 

 only by the nature of the differential equation (122) 

 and by the two boundary conditions, namely that the 

 sound pressure is zero at the surface and that no 

 soimd is coming up from below the projector. Thus 

 the values of Ki are independent of the frequency, 

 sound velocity, and velocity gradient. 



Once these characteristic values of K have been 

 foimd, the corresponding values of m to be used in 

 equations (115) and (116) can be found directly. By 

 substitution in equations (123) and (119), we find 



K J ^J r^ (124) 



2 

 Mi 



^ 



cl 



Ki. 



The second term in equation (124) is always very 

 much less than the first in cases of practical im- 

 portance. Even for a temperature gradient as large 

 as 1 F per ft of depth increase, and for a frequency 

 of only 100 c, the second term is less than 1 per cent 

 of the first for Ki less than 10, the region of practical 

 interest. Thus we may take the square root of equa- 

 tion (124), expand in a series, and retain only the 

 first two terms. This process gives 



27rr KifBcAil 



27r Ki/irfBAi 



Let Ki be the characteristic value of K with the 

 smallest imaginary part, and let this imaginary part 

 be denoted by iK'i. Let the theoretical sound pressure 

 associated with the characteristic value Ki be pi. In 

 the shadow zone the intensity is proportional to the 

 square of pi since the soimd pressures associated with 

 the other characteristic values K,- may be neglected. 



The intensity level found from equation (119) is 



L = 20 log pi = C - 20(logio e)^( — ) 'x (126) 



4 \ Co / 



where C includes Ai and the other variables taken 

 over from equation (118). While C changes gradually 

 with position, it is nearly constant along the shadow 

 boundary. Multiplying out terms in equation (126), 

 and using equation (111) for B, we get, finally, 

 5MK[f'i-dc/dy)'x 



L = C 



Co 



(127) 



It should be emphasized that equations (126) and 

 (127) apply only in the shadow zone. In the main 

 beam other terms corresponding to other values of 

 Ki must be considered. 



The analysis in a report by Columbia University 

 Division of War Research' considers the radiation in 

 three dimensions sent out by a point source and is 

 thus more general than the simple analysis presented 

 here. However, the final result for the sound in the 

 shadow zone is nearly identical with equation (127); 

 the only difference is that the term b.QbK[ becomes 

 25.7 in the exact computation of reference 7. With 

 this substitution, we have the following formula for a, 

 the attenuation coefficient beyond the shadow bound- 

 ary in decibels per unit distance. 

 2?>.7f\-dc/dy) 



a = 



Co 



(128) 



In this equation /is the sound frequency in cycles per 

 second, and dc/dy is the velocity gradient in feet per 

 second per foot. If Co is in feet per second, formula 

 (128) gives the attenuation in decibels per foot; if Co is 

 in yards per second, the result is the attenuation in 

 decibels per yard. 



Since inverse-square spreading is quite negligible 

 compared to the intensity drop at the shadow 

 boundary, equation (128) gives the slope of the 

 transmission anomaly at points beyond the shadow 

 boundary. However, this equation cannot be used 

 at shorter ranges and must therefore be regarded as 

 an expression for the local attenuation coefficient in 

 the shadow zone. 



Equation (128) is compared with observational 

 data imder Attenuation Coefficient at Shadow Bound- 

 ary in Section 5.4.1, where it is shown that the ob- 

 served local attenuation coefficients beyond the 

 shadow boundary are not more than about half the 

 predicted values. In other words, in practice much 

 more sound appears ui the shadow zone than is pre- 

 dicted by equation (128). 



