Impulsively Generated Waves Propagating into Shallow Water 



The Bessel functions Jo(z) and J|(z) are computed from series 

 expansions given in Abramowitz and Stegun [ 1964] (eqs. 9.41-9.46), 

 J2(z) and J3(z) are then computed from the recursion relation 



Jn-|(2) + Jn.|(z) = (2n/z)Jn(z). 



The time of arrival of each frequency at the starting station 

 is computed from the relation , 



tijK=V^- (29) 



ijk 



It is now possible to compute the wave spectral energy at the 

 starting station by applying Eq. (12). That equation, rewritten for 

 numerical integration, is: 



r\pg Ijk Ijk^jk 



where 



a - y J<mifi. COS 9o,n Zss' .^ 



Ijk Z^ /Cij^, cos 9;^^^ cos e^.i^j^^ ' 



^ijk = ^I'j Z 4 (t^^) 



As' 



2V-' cose,.,.,, 



2 



1 J_ Q-jj Q-Ji (1-2 cosh 2crii ) 



= v' yr ^ 1 .[" 4 " sinh 2o-i| " sinh'^ 2o- 1 



^Ij Z_y Lg tanh o-j. J *L Z' i + _j£li__ ^ ^ J 



'=0 V sinh 2(T-j. / 



. As' (32) 



* ^'^^ ^i-l,j,k 



The last factor in Eqs. (31) and (32) represents the incremental 

 distance along the path where As' is the station spacing on the axis. 



The energy may now be carried forward from station to 

 station as : 



i+l,J,k ijk 



or 



265 



