PORTER: SOFAR PROPAGATION OF WIDE-BAND SIGNALS TO LONG RANGES 



F(a)) is the spectrimi of the transmitted pulse or shot, A is a weiqht- 



m 



ing term containing the cylindrical spreading factor for each mode as 



well as some normalization terms, and the exponent contains the 



horizontal propagation constant for each mode. is the angle the 



m 



propagation vector makes with channel axis. The eigenf unction $ is 



found from the WKB approximation where 9 is the phase integral from 



depth z to the turning point z . y is the vertical propagation 



constant. The cos term expresses the mode interference and its 



dependence on z. It is zero when the argument is (2n + 1) 17/2 and 



largest when the argument is mir. The acoustic field is found by 



integrating the normal mode expansion over all w. The interference 



terms from the eigenfunctions are lumped together into an interference 



factor M; remaining terms have been placed in B . 



m 



The integral over frequency is evaluated by the method of 

 stationary phase. The group velocity of speed at which energy of 

 frequency w and mode m travels is found by minimizing the exponent 

 of the propagation integral subject to the constraint that the charac- 

 teristic equation or resonance condition for the normal modes be 

 obeyed. 



The resonance condition is simply the requirement that the phase 

 of a mode vary by mir between the upper and lower turning points . The 

 dispersion curve of relationship between frequency and group velocity 

 is found by combining the first two equations in Figure 8. In order 

 that the major contribution to the phase integral comes from the 

 vicinity of the stationary phase point, the product of the time spread 

 of the channel and the bandwidth of the pulse must be large. 



The next-to-last equation in Figure 8 results from evaluating 

 the propagation integral at the stationary phase point. It is clearly 

 a summation over all modes with each mode weighted by the interference 



647 



