STICKLER: NORMAL MODES IN OCEAN ACOUSTICS 



and the density by a constant. He sums the proper modes and evaluates 

 the EJP branch integral. 



Examples 



In this section two comparisons are made; they are chosen to 

 illustrate the importance of the continuous modal contribution. 

 Consider the profile shown in Figure 8; it is of the type considered 

 in Figure 1. It is very interesting because, at 50 Hz, there is only 

 one proper mode, and it is quite near cut-off. In Figure 9, the 

 transmission loss is shown at 50 Hz for a source at a depth of 20 feet 

 and a receiver at 40 feet. The lower solid curve represents the 

 contribution of the single proper mode. Blatstein's calculation (see 

 Spofford, 1973) for this one proper mode is in good agreement. Bart- 

 berger (1973) has summed not only the one proper mode but several 

 of the improper modes. However, for this case, it is seen that the 

 leaky modes make virtually no contribution. Bartberger's calculation 

 does not include the corresponding Pekeris-type branch. The upper 

 solid curve is the sum of the one proper mode plus the EJP brainch as 

 calculated by Stickler (1975) . The results of Kutschale (1970) , who 

 sums the proper modes and adds the EJP branch contribution, are seen 

 to be in close agreement. 



This calculation shows two interesting points: 1) The contribu- 

 tion of the continuous modes can be important to many water depths, 

 and 2) the sum of the leaky modes is not always a good approximation 

 to the EJP branch integral. 



Figure 10 shows a plot of transmission loss for the same geometry 

 except now the frequency is 100 Hz. There is still only one proper 

 mode, the smooth lower curve. The upper solid curve shows the contri- 

 bution of the proper plus the EJP branch contribution, and the dots 



146 



