street^ Chan and Fromm 



and Mogel, et al. [1970] . These methods, based on linear, geo- 

 metric-optics theory, are applicable to arbitrary bottom topography 

 but can predict neither breaking nor run-up. They also neglect 

 reflection and diffraction effects. The computer code is very simple, 

 requiring step-by- step solution of Snell's law and a wave intensity 

 equation over a grid of bottom depths. 



Vastano and Reid [ 1967] described a procedure employing a 

 numerical integration of the linearized long wave equation to study 

 tsunami response at islands. They concluded, by comparison with 

 analytic solutions for special cases, that their numerical model gave 

 an adequate representation of the solution to the linearized equations. 

 Their paper indicated that the work was a lead-in to a more general 

 treatment of arbitrary bottom topography. At the island they used a 

 vertical cylinder that penetrates the surface and thereby restricts 

 the movement of the instantaneous shoreline. No run-up can be cal- 

 culated in the usual sense. 



In another approach, Lautenbacher [ 1970] used linear, 

 shallow water theory to study run-up and refraction of oscillating 

 waves of tsunami-like character on islands. His method allows for 

 the moving, instantaneous shoreline and, of course, for superposi- 

 tion of individual results from monochromatic waves. Working from 

 an integral equation formulation and employing a Hankel function 

 representation of the far-field radiation condition similar to that 

 used by Vastano and Reid [ 1967] , Lautenbacher used a grid of dis- 

 crete points to numerically integrate the integral equation of his 

 model. Combining his work with Carrier [ 1966] , Lautenbacher was 

 able to estimate total tsunami run-up from a distant source. He also 

 emphasized the importance of refractive focusing effects. 



The numerical methods aimed specifically at modelling non- 

 linear effects take three forms: 



a. Approximate, plane-flow models for arbitrary or sloping 

 beaches and based on approximate equations. 



b. Exact plane-flow models. 



c. Quasi- three- dimensional models. 



The approximate, plane-flow models are represented by the 

 work of Freeman and LeMehaute [ 1964] , Peregrine [ 1967] , Heitner 

 [ 1969] , Street, et al. [ 1969] , Camfield and Street [ 1969] , and Mads en 

 and Mei [ 1969a, 19^9b] . With the exception of Heitner [ 1969] , the 

 authors used Eulerian coordinates. Freeman and LeMehaute [ 1964] 

 applied the method of characteristics to the nonlinear, shallow water 

 wave equations for plane flow. They described a method for com- 

 puting the shoaling of a limit- height solitary wave on a plane beach, 

 predicting the point of breaking inception by the crossing of character- 

 istic lines and computing the subsequent bore development and run-up 

 at the shoreline. A term was added to the equations to correct the 



148 



