H = -101og 



N 

 .2 



PsPh ^ X "O ^^'^ '^n(z) Un(zo) 



n=l 



2 

 + ocpj, (2) 



2 

 where r is the range, zq is the source depth, z is the receiver depth, Hq is the Hankel function 



of order zero, second type, X^ is the nth eigenvalue, Uj^ is the depth function for mode n, 

 and Pg and p^ are the densities at source and receiver. The sum is over the number of modes, 

 N, making a significant contribution. The final term contains the volume attenuation coeffi- 

 cient, a\. From Thorp (ref 6), ocx in dB/m is computed by the relationship 



0.9144 a^ = 0.0001 F^/d + F^) + 0.04 f2/(4100 + F^), (3) 



where F is the frequency in kHz. Improved equations or those for specific ocean areas can 

 be easily substituted. The depth function, U^,, is a solution to the depth-dependent part of 

 the separated wave equation 



d^U/dz^ + [gj2/c2(z) - X^] U = 0, (4) 



where 



to = 27rf 

 and f is the frequency, in Hz. 



A closed-form solution to eq (4) can be obtained when the reciprocal sound speed 

 squared or squared index of refraction is a linear function of depth. That form is used in 

 this program, and sound speed in each layer is expressed as follows: 



[Ci/c(z)]2=l-27i(z-Zj)/ci, (5) 



where Cj, Zj, and y[ are the sound speed, depth, and sound-speed gradient, respectively, at the 

 top of layer i. Up to 12 such layers are permitted by the program, for modeling the sound- 

 speed profile. 



With this expression for sound speed, solutions to eq (4) can be expressed in terms 

 of solutions to Stokes' equation 



h" + zh - 0. (6) 



Only a simple change in independent variable is required from z to f , where 



fjCz) = [af (z - Zj) + a;2/c2 _ ^2] /a? (7) 



and 



af = -27iCo2/c^^. (8) 



6. Analytic Description of the Low-Frequency Attenuation Coefficient, by WH Thorp; J Acoust Soc Am, 

 vol 42, p 270, 1967. 



