Impulsively Generated Waves Propagating into Shallow Water 



field point desired. Treating E((jo) as a constant a new r[\ = T|(a)) 

 will be computed. Details will be discussed in the appendix on com- 

 putational procedure. 



The phase of the waves in water of variable depth requires 

 attention. The phase function of Eq. (4), cos (<rr - Qt) , for uniform 

 depth may be rewritten as 



cos 



(^ -V-) 



since t' = r'/v'« Now, however, in water of variable depth wave 

 number and group velocity will depend on location as well as fre- 

 quency. The argument must now be represented in integral form 

 with the integration performed along the path, s. The phase term 

 now becomes 



cos 



[y^(.-^)ds-] (iz) 



where K = /f(h,w), v' = v'(h,a)), and h = h(s'). 



The central assumptions involved throughout this development 

 have been that the system is linear and conservative and that the 

 energy per unit frequency remains constant. The assumptions are 

 quite viable as long as the water is of deep to moderate depths, say 

 one-half wave length or deeper. Inshore of this point the system 

 becomes progressively more non-linear, non- conservative and the 

 frequency assumption more vulnerable. An evaluation of the linear 

 assumption will appear in the section. 



IV, NON-LINEAR FEATURES OF THE SHALLOW WATER SYSTEM 



The previous section carried the wave system from a region 

 of uniform depth into a region of variable depth; however, the des- 

 cription of the system retained Its linear character. We have des- 

 cribed earlier the change In form of shallow-water waves from one 

 of sinusoidal fornn to one of sharp crests separated by long flat 

 troughs with an associated horizontal plane asymmetry. In this 

 section a particular non-linear theory, the cnoldal wave theory of 

 Keulegan and Patterson [ 1940] , will be Incorporated to modify the 

 form of the waves and the wave envelope. 



The first of the cnoldal family of wave theories was presented 

 by Korteweg and de Vrles [ 1895] , Because the wave elevation was 

 given In terms of the Jacoblan elliptic en function they coined the 

 work cnoldal to describe the resulting wave formi. Subsequent contrl- 

 butlons , In addition to the Keulegan and Patterson paper cited above. 



249 



