Note that a value of v that makes this determinant zero, or near zero, ordinarily is 

 zero because only one diagonal element is very small. For trapped modes this element is at 

 the row representing the first interface below the mode, ie the interface just below the layer 

 of positive gradient in which the sound speed is equal to the mode phase velocity. For 

 unstrapped modes it is usually the final diagonal element that is small. Thus the layers in 

 which the sound speed is greater than the phase velocity of a mode do not greatly affect the 

 eigenvalue. Eigenvalues are determined mainly by those parts of the sound-speed profile 

 that are less than the phase velocity. 



When an eigenvalue is found, the coefficients Aj and Bj must next be evaluated. As 

 mentioned earlier, one coefficient can be arbitrarily chosen. This is done, and eq (21) is satis- 

 fied by letting 



and 



Ai=pih2[ri(0)] 



Bi=-|Oihi[fi(0)]. (24) 



The factor p j is used simply because the number containing it is easily available in the pro- 

 gram. It is divided out by the normalizing factor, D. Eq (22) and (23) can then be used to 

 evaluate the remaining coefficients, but the triangularized form of the matrix yields the coef- 

 ficients with less computation. If gy is the element in the ith row and jth column of the tri- 

 angularized matrix, then by Cramer's rule, 



Bi = Ai_ig2i_2,2i-2g2i-l,2i/Ei 

 and 



Aj = -Ai_i g2i-2,2i-2 g2i-l,2i-l/^i' 

 where 



Ei = §21-2,21-1 g2i-l,2i- §21-2,21 §21-1,21-1- (25) 



A simpler form is used for B^ in the final layer since there is no A-^ there. 



In certain situations numerical problems can arise in evaluating the determinant. 

 These require some extra tests in the subroutine that makes the evaluation. The extra tests 

 will be discussed in the section, NUMERICAL BREAKDOWN. A more routine problem is 

 the loss of accuracy that can arise in subtractions in the row reduction of the matrix. This 

 loss results in less sharpness of convergence to a root. The size of the determinant, G, can be 

 14 orders of magnitude less at a root than at the general background near the root. This 

 variation occurs because the modified Hankel functions can be computed to about 14-place 

 accuracy in a computer with 18 decimal places available. Modes usually converge to 10 or 

 12 places; thus a few places are lost in evaluating the determinant. In some profiles, usually 

 those with multiple ducts or those in which propagation through bottom sediments plays a 

 large part, the convergence can be much poorer. Modes need to converge to about 4 places 

 to be reUable for computing losses, and convergence occasionally fails to meet this require- 

 ment. The only current cure for this loss in accuracy is to go to higher-precision arithmetic 

 or to compute the modified Hankel functions to greater accuracy. For instance, a standard 



