CHARACTERISTIC VALUES FOK THE BILINEAR M CURVE 



229 



FIGURE 2. First standard and first trapped mode. 



In discussing the behavior of U in relation to the 

 Y curve, it is best to plot U or \ U \ rather than 

 decibels. It is also helpful to draw a vertical line at 

 the abscissa —B, and this is then usually used as 

 the axis in plotting U or \ U \. The diagram for a 

 trapped mode shows that | U \ shows exponential 

 decay in the "barrier" region where the Y curve lies 

 to the left of the line at —B. Below the barrier U 

 shows oscillatory behavior, but with no nodes for 

 the first mode; above the barrier U is a spiral, which 

 shows only as a slow increase in the plot of | U \. 



This same difference in the behavior of U for Y > 



or < — B is a useful tiling to remember in more 

 general cases. Sometimes it is not quite so clear-cut 

 as in this case of trapping. If A is not small, the 

 situation cannot be so completely defined in terms 

 of B alone. It is certain, however, that | U | will 

 show essentially exponential behavior in any region 

 where the "height of the barrier," i.e., the amount 

 by which the Y curve lies to the left of the line at 

 — B, is greater than A. 



This sort of general physical consideration about 

 the U curve leads, on being put in more precise 

 mathematical form, to the phase integral methods. 

 Unfortunately, no phase integral method can claim 

 validity for this model except in cases of trapping. 

 In general, the Eckersley phase integral method for 

 untrapped modes requires that the Y curve be an 

 analytic function, and the bilinear curve obviously 

 is not. Most of the values presented are, accordingly, 

 the results of exact calculation. 



Figure 3 shows that for negative s the attenuation 

 falls rather suddenly to very small values at a certain 

 value of g and then quickly approaches zero. This 

 indicates the occurrence of trapping. On the other 



Figure 3. Attenuation constant versus anomaly height for bilinear M curve, first mode. 



