Impulsively Generated Waves Propagating into Shallow Water 



2 -4 



Now, smaller roots of g(k) , say < k ^ 1 - 10 , are 

 readily obtained by iteratively searching for zeros of g(k) in 

 successively finer increments. Larger roots of g(k) , say 

 1-10" <k:21-10' , are then obtained by Iteratively searching 

 for roots of g(k) in half-power increments of 10'", where k^ = 

 i - 10' . Nearly exact solutions in terms of n are then obtained 

 using the approximate interpolation relation 



n = a/«°-^^^+P (36) 



where a and P are interpolation constants and J2 is the dimension- 

 less frequency. This approximation is based on a family of curves 

 (n vs. t) in Wiegel [ 1964] (Fig. 2.24). 



Computer computational difficulties are avoided in solving 

 Eq, (18) for values of k near unity, since the modulus k and the 

 complete elliptic integrals K(k) , E(k) can be determined from the 

 value of 10'" = 1 - k , using the approximations given in Abramowitz 

 and Stegun [ 1964] (eqs. 17.3.33-17.3.36). Since only the largest 

 real root of g(k) is of interest, the computer program searches 

 first for the largest real root. If no real root is found in the range 

 1 - 10'^ < k < 1, then the largest real root in the range < k^ 

 1 - 10" is computed. The smaller roots or imaginary roots have 

 no meaning. Whenever no real root is obtained in the range 

 < k < 1 , the modulus Is set equal to zero. The computation is 

 repeated for each frequency at each location. Note that the calcula- 

 tion Is based on the double amplitude of the envelope and not on the 

 phase wave elevation. 



The distortion of the envelope to Its asymmetrical form Is 

 achieved by applying Eq. (16). Denoting the elevation of the envelope 

 above and below the SWL as HI and H2 those equations become 





krjkK(ki^,) 



(37) 



H2ijk - Hijk[i - -nrif-J 



The phase term for water of variable depth was given as 

 Expression (13). Denoting the argument of the function as 4^ and 

 the field point on s as Sj 



,= j'J'(.-^)ds'. 



267 



