TWO METHODS FOR THE COMPUTATION OF 



THE MOTION OF LONG WATER WAVES - 



A REVIEW AND APPLICATIONS 



Robert L. Street 

 Robert K. C. Chan 



Stanford University 



Stanford^ California 



and 



Jacob E. Fromm 



IBM Corporation 



San JosBj California 



I. INTRODUCTION 



The continuing evolution in speed and capacity of digital com- 

 puters has encouraged the development of many computationally 

 oriented methods for analysis of the movement of waves over the 

 surface of the ocean and onto the shore. Carrier [ 1966] gave analy- 

 tical techniques requiring numerical evaluation for the propagation 

 of tsunamis over the deep ocean and for the run-up on a sloping beach 

 of periodic waves that do not break. He noted that linear theory is 

 valid in the deep ocean and over much of the sloping shelf; thus , non- 

 linear theory is needed only in specific regions where the nonlinear 

 contributions to the dynamics are important. However, his nonlinear, 

 approximate theory was developed only for the plane flows. An ex- 

 tension and application of Carrier was made by Hwang, et al. [ 1969] . 

 They studied the transformation of non-periodic wave trains on a 

 uniformly sloping beach using the nonlinear shallow water wave 

 equation and the Carrier-Greenspan transform. This transform 

 fixes the moving, instantaneous shoreline of the physical plane to a 

 single point in the transformed plane. Although the analysis deals 

 only with plane flows and does not handle breaking waves , it does 

 predict wave run-up and reveals a significant beat phenomenon. 



To study nonlinear effects and/or to account more completely 

 for the waves' reaction to arbitrary ocean topography and boundaries, 

 it is natural to turn to numerical methods and their accompanying 

 computer codes. The simplest of the numerical methods are repre- 

 sented by the refraction techniques of Keulegan and Harrison [ 1970] 



147 



