CONVERGENCE ZONE DEPENDENCE ON FREQUENCY 



R. M. Fitzgerald 



Naval Research Laboratory, 

 Washington, D. C. 



This presentation concerns a comparison of transmission- 

 loss curves for 13.89 Hz and 111.1 Hz. Because of the 

 source-receiver geometry and the ocean environment of the 

 experiment, the predominant modes of propagation were RSR- 

 type paths. Thus, we are considering frequency effects and 

 the formation of convergence zones. Transmission-loss cal- 

 culations based on modal solutions to the wave equation 

 compare favorably with measurements. 



For a range- and azimuth -independent medium the wave equation is 



separable. At long ranges its solution can be approximated by a 



finite sum of discrete normal modes. Using the asymptotic form of 



the Hankel function, one obtains the solution for the acoustic pressure 



given in Figure 1. The reciprocal square root of range results from 



geometric spreading. The mode summation describes the interference 



effects. The quantity (jj is the source frequency and K is a separa- 



m 



tion constant. The amplitudes P depend on the source and receiver 



m 



depths and indicate the extent to which a mode is excited and 

 detected. 



The phase velocity, C , of the m mode is defined as oj divided 



m 



by K . It is useful for a qualitative description of the eigenfunc- 

 m 



tions. The depths at which the sound speed in the medium equals the 



m phase velocity, C , correspond to the turning points of the m 



m 



eigenf unction. Where the speed of sound is less than the phase 

 velocity of an excited mode, the eigenf unction is oscillatory and 

 the energy can be concentrated. Where the speed of sound exceeds 

 the phase velocity, the eigenf unction is exponentially decreasing. 



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