where 



z^ is a suitably chosen reference point 



z^ is a turning point 



Bi is the Airy function of the second kind. 



Whenever w ^■^p is a moderate to large negative number, F+ reduces to the 

 usual WKBJ representation ~ 



F+ (z;X,w) = |c"2 (z) - X^l exp|+ iwQ (Zq, z;X)}' . 



Since this modified representation is inaccurate in the vicinity of double turning 

 points, F+ are arbitrarily set to zero whenever they occur. Hopefully, this 

 will only affect a small interval of integration and will not introduce significant 

 errors in the final result. Murphy's technique offers an alternative procedure. 



The method of evaluating the ray type integrals is based on the following: 



1. Segment the interval of integration into suitably chosen subintervals. 



2. Use stationary phase to evaluate subintegrals whenever possible. 



3. Integrate the remaining cases numerically. 



It was originally though that the numerical integration, although lengthy 

 when compared with stationary phase, would be invoked so infrequently that its 

 contribution to the total computer running time would be inconsequential. So far, 

 this has not been the case. Hopefully, the running time will be reduced eventually 

 when the integration routine is made more efficient. 



Since it is customary to give computer programs names so that they may 

 be distinguished from others performing similar functions, the program used 

 herein is called CONGRATS V, where CONGRATS is an acronym for Continuous 

 Gradient Ray Tracing System. Actually, the completed program will predict the 

 performance of active and passive sonar systems and is, therefore, more than 

 just a propagation program. As shown in figures 1 through 3, CONGRATS V has 

 the option to invoke several ray tracing procedures. The propagation losses were 

 obtained by adding the multipath contributions coherently. CONGRATS V also 

 produces a plot of propagation loss using power addition, in which case the 

 phases of the individual contributors are neglected. 



108 



