The standard correction to mode 1 is shown in figure 2 by the dot-dashed line. In the 

 depth interval where f < -8.4, the function is set to zero. The values of the depth function 

 at greater depths result from a modification in the values used in the determinant. 



The corrected values in figure 2 are not equal to the true value of the depth function, 

 but they are small enough that they do not alter the propagation loss to a tenth of a decibel 

 when a full set of modes is used. When the source or receiver is at a depth where such cor- 

 rections are necessary, the mode can be omitted from the computation. Thus, properly 

 omitting modes would solve the above problems except for the cases where the normalizing 

 factor, D, is affected. In these cases, losses cannot be computed accurately without the 

 corrections. 



PROGRAM MODIFICATIONS 



The modification is approximately equivalent to modifying the sound speed profile 

 as shown by the broken line in figure 1 . In effect, the sound speed is not allowed to become 

 enough greater than the phase velocity of the mode being considered to cause problems. 



The limitation on f is accompUshed at three different places in the normal mode pro- 

 gram. It is not clear that this is the best way to handle the problem and it may be redundant, 

 but it appears to be an adequate solution. These three corrections will be described next. 

 Finally a correction to the determinant program is described which is necessary because the 

 limiting of ^ can cause false zeroes in the determinant. 



In the subroutine SETUP the elements of the determinant are computed by deter- 

 mining f at the top and bottom of each layer and then calling the modified Hankel function 

 program. At the top of each layer, Re f is set to -7.5 if its value was less. However, this is 

 done in an iterative loop in which the real part of cj/Cj in eq (7) takes on a sufficiently larger 

 value while its imaginary part is fixed. This is done to retain the absorptive properties of a 

 layer when its sound speed is in effect being reduced. It has been found unnecessary to make 

 the above constant, -7.5, a function of T-lim which the user can vary, because an oversized 

 value at the top of a layer is not as critical as at the bottom. At the bottom of a layer, several 

 tests are made. If the real part of f has decreased past the limit at some depth between the 

 top and bottom of the layer, it is set at the limit. This Umit, called S-lim in the program, is 

 related to T-lim by the relationship 



S = -(T)2/3 (42) 



where S and T are the two Umits. If Re f is less than -7.5 throughout the layer, it is simply 

 set at -7.5. Such a layer has negligible effect on a mode. 



In program MAIN at the location where depth functions are computed for given 

 depths, a process similar to that above is used. To evaluate the depth function in a given 

 layer, f is first evaluated at the top of the layer. The real part of f is theij limited as in the 

 program SETUP above. Next f is evaluated at the given depth by adding the depth-dependent 

 part onto the value at the top of the layer which may be the modified value. If this final 

 value is less than S-lim, the depth function is set to zero. If it is greater than S-lim, the func- 

 tion is computed in the usual way. 



