FINDING EIGENVALUES deals with the philosophy of eigenvalue location employed 

 by this program, which essentially leaves this function to the user, the program only serving as 

 a tool. It shows how the program is used to make computations. 



Several "automatic" mode finding versions of this program have been developed to 

 the point of accommodating certain classes of profiles. However, they need further develop- 

 ment and have not yet been reported. 



SOUND SPEED PROFILE indicates the required equations for curve fitting and the 

 various ways the sound speed can be read in on cards. A continuous water profile can be 

 entered quite simply, but sediment layers with sound speed discontinuities and absorption 

 gradients can become complicated. 



REFLECTION COEFFICIENTS AND OTHER AUXILIARY OUTPUTS describes a 

 short subroutine that computes reflection coefficients for any mode at a given profile inter- 

 face. Intermode interference lengths and mode damping coefficients are also discussed. 



COMPUTATION OF THE MODIFIED HANKEL FUNCTIONS gives the analysis 

 necessary for computing these functions. The use of continued fractions to evaluate an 

 asymptotic series is discussed. To facilitate running the program on computers of different 

 word length, this section provides the information required to optimize the functions for 

 the different word lengths. 



MODE FOLLOWER PROGRAM describes a separate but related program for investigating 

 the eigenvalues themselves rather than using them to compute propagation losses. 



GENERAL SOLUTION 



The derivation of the normal-mode solution has been discussed from various points 

 of view (eg ref 1 , 4, 5). Only an outline is given here. In general, the time-independent wave 

 equation is written in polar coordinates and the azimuthal coordinate is dropped under the 

 assumption that the field is independent of azimuthal direction. Thus 



(1/r) 0/9r) [rO^/ar)] + (d^^/dzh + (co^/ch ^ = 0, (1) 



where i// is the velocity potential, c the sound speed, and the independent variables are depth, 

 z, and range, r. 



Equation (1) is then separated into range- and depth-dependent parts with a separation 

 constant X. The separation is possible when the sound speed is a function of depth only. 

 After accounting for the source discontinuity and the outgoing radiation condition, integrating 

 over all real values of the separation constant, and normalizing, one can find the solution for 

 a field point in terms of propagation loss H as follows; 



4. Naval Air Development Center Report NADC-72002-AE, Normal Mode Solutions and Computer Programs 

 for Underwater Sound Propagation, by CL Bartberger and LL Ackler, 4 April 1973. 



5. A Normal Mode Theory of an Underwater Acoustic Duct by Means of Green's Functions, by RL 

 Deavenport; Radio Sci, vol 1, p 709-724, 1966. 



