

f 



58 









^ 



^ 



^N' 



55 

 \ 



56 

 \ 



57 





1 



7\ 



^ 



^ 



V 59 



\[\ 



y 



1 



n 



V 



56 



57 



58 



50 51 



y 



52 



53 



54 



55 56V5 



58 



1 



1 



y^SD 









1 



1 





1 



1150 1200 1250 1300 1350 1400 1450 

 FREQUENCY, Hz 



Figure 9. The imaginary parts of the modes whose 

 real parts are shown in figure 8 expressed as mode 

 attenuation. To avoid confusion, they are not 

 shown across the full frequency interval. 



When frequency is the variable parameter, the group velocity of the mode can be 

 computed easily since a numerical derivative can be computed. Group velocity is given by 

 the relationship 



Cg = dco/d (Re k) 

 = Aw/A (Re k) 



s -Afv^/f Av , 



where k is the horizontal wave number of the mode and v is the real phase velocity. The 

 mode follower prints this value out at each step, along with the eigenvalue. 



IMPLEMENTATION OF THE MODE FOLLOWER 



The mode follower was originally implemented for a two-layer normal mode which 

 differed from the n-layer program in that the derivative dG/dv of the characteristic equation 

 was evaluated along with G. The iteration for roots of G was thus Newton-Raphson and is 

 given by the relationship 



Vi+i = Vj-G/G' 



(86) 



This is simpler than the secant iteration of eq (15), in which G' must be evaluated numeri- 

 cally. Because of the simpler iteration, an effective scheme for mode following was available. 



38 



