modified Hankel functions or to the argument of the exponential function within modified 

 Hankel functions. A T-lim value of 25.0 is used in the program, but a smaller number occa- 

 sionally is required. The program user can alter T-lim by appropriate input cards (Key 8 - 1 

 followed by a new value of T-lim). The next few paragraphs demonstrate the symptoms of 

 this problem, so as to assist a user in recognizing the problem. The remainder of this section 

 describes the modifications that have been made to the computer program to correct this 

 loss of accuracy. 



The solid line of figure 1 shows a simple surface duct and the phase velocities of the 

 first three modes at 3 kHz. For this profile, the depth function of mode 1 is shown in fig- 

 ure 2. The solid Hne is the depth function as computed by a program that does not correct 

 for numerical breakdown. The dashed line shows the correct depth function below a depth 

 of 71 m. This result was determined from Airy functions, not from the program. Between 

 depths of 71 to 100 m, the program cannot compute the depth function accurately. In the 

 second layer, which starts at a depth of 100 m, the function can be computed accurately but 

 it is incorrectly placed by the boundary condition that requires the depth functions to be 

 continuous at interfaces. The slope of the depth function was correctly computed as indi- 

 cated by the identical shape of the three depth functions in the second layer. The shape is 

 such as to make the correct depth function continuous in slope across the interface. 



The breakdown in accuracy at a depth of 71 m occurred when f had a value of -8.4. 

 (f is given by eq (7) and is the argument of the modified Hankel functions.) A negative value 

 of f occurs when the mode phase velocity is less than the speed of sound. Since the ray of 

 the same phase velocity cannot reach such a region, the sound field there is a diffracted field. 

 The mode depth function is therefore small at such depths. In the figure, the depth func- 

 tion amplitude at the breakdown point is about 7 orders of magnitude (or in terms of propa- 

 gation loss, 140 dB) down from its maximum. Equations (62), (66), and (68), which will be 

 given for the modified Hankel functions, indicate that the argument of the exponential term 

 is 2/3(8.4)3/2, or 16.2. The functions hj and hj will thus be about 10^ in magnitude at a 

 depth of 71 m. These large values and their small difference account for the approximate 

 accuracy loss of 14 decimal places, which is the general accuracy of the modified Hankel 

 functions. 



Incorrect behavior in the depth function usually occurs when f is about -8.4. In 

 some more compUcated profiles, however, where accuracy is lost in row reduction of the 

 determinant, the depth functions may become incorrect at values of f that are less in abso- 

 lute value. When this problem occurs it can be diagnosed by plotting the depth function of 

 the mode and noting the steep positive slope through some depth interval as in figure 2. 

 When that occurs, the value of T-hm should be decreased. 



Incorrect depth functions can cause errors in propagation loss computations in two 

 ways. In figure 2, the soUd-hne depth function, because of its large size, can cause losses to 

 be too low at a depth of around 1 00 m. The second error would occur if the duct were 

 deeper, say 110 m. At this depth the erroneous segment of depth function in figure 2 would 

 reach a value of about 10"^, where it would be larger than the correct lobe of the depth 

 function near the surface. With this extra area under the curve, the normalizing factor would 

 be increased significantly and would reduce the size of this entire depth function. Thus, 

 losses near the surface would be larger because of the loss in size of mode 1 . 



16 



