Van Mater and Neat 



I. INTRODUCTION 



The nature of wave systems generated impulsively by explo- 

 sions at or below the water surface is of natural interest to re- 

 searchers in the naval community because of the ship behavior which 

 results from such an environment. When water-wave systems enter 

 shallow water they undergo changes in form which may have an 

 adverse effect on motions depending on the size of the waves relative 

 to the ship or small craft. While the motivation of this work from 

 our point of view is ultimately the prediction of ship behavior in a 

 shallow-water explosion- gene rated wave environment, this paper 

 is confined to the prediction of the forcing function -- the wave 

 system. In itself this case presents an interesting means of study- 

 ing the shoaling behavior of a dispersive wave system, an area 

 which has received surprisingly little attention. Previous efforts 

 in the direction of predicting impulsively- generated wave systems 

 entering water of variable depth stem from the work of Dr. William 

 Van Dorn (cf. Van Dorn and Montgomery [ 1963] ) and have been 

 confined to the prediction of wave envelopes. This paper should be 

 viewed as a second generation of the Van Dorn model. 



As a starting point we shall tabulate some of the rather com- 

 plex effects which occur in shallow water. Not all of these will be 

 considered in the present prediction scheme but the exclusions will 

 be noted. 



(a) In deep water, phase and group velocities depend pri- 

 marily on wave frequency giving rise to the well-known 

 characteristic of the system known as frequency disper- 

 sion. As the system moves into water whose depth is 

 small compared to the lengths of the waves in the system 

 this frequency dependence weakens and dependence on 

 water depth and wave height strengthens. Waves which 

 in deep water moved through the group at phase velocities 

 up to twice the group velocity now become nearly frozen 

 in their position in the group. 



(b) The nearly sinusoidal form of the waves in deep water 

 changes to one of sharp crests separated by long flat 

 troughs. An asymmetry about the horizontal plane 

 develops in which the crest height above the still water 

 line is greater than the trough depth. The maximum 

 slope of the waves increases. This last feature is of 

 particular importance in ship motion prediction, 



(c) As wave height becomes significant with respect to the 

 water depth and the wave nears the breaking point the 

 leading face of the wave steepens and a wave slope asym- 

 metry develops. As the slope of the face of the wave 

 near the crest approaches the vertical the wave becomes 



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