V * = ®T&- 



x 



max 



and 



n = I 11 (67) 



x 3x v J 



This is discussed by Peregrine (1970) . He points out that nonlinear 

 terms which were neglected in the linear equations (57) and (58) cause 

 a cumulative error that may become appreciable in a numerical solution 

 after a time given by 



H3/2 



(68) 



H g 



1/2 



Where rapid shoaling occurs, i.e., where a wave passes over a large 

 change in water depth in a relatively short period of time, the accumu- 

 lated error will be much smaller than for slow shoaling, where the wave 

 passes over the same change in water depth in a relatively long period 

 of time. 



The finite-amplitude equations (59) and (60) are valid as long as 



U* > 1 



but generally become invalid after a finite time as the front face of 

 the wave steepens. The Boussinesq equations are also applicable where 

 U* > 1; i.e., where {j\ x ^max < (H/d). Peregrine (1970) points out that 

 the Boussinesq approximation works well for values of H/d up to about 

 0.5. The Boussinesq or the Korteweg-deVries equations are used for 

 waves approaching a shoreline where values of H/d become large. 



Hammack (1973) gives the value of U* as 



— L_ (69) 



max 



u* = (I!™*)' 



to describe a particular region of a complex waveform. However, the 

 value of U* would be expected to vary from region to region of the 

 waveform in this case. This variation would indicate that using a single 

 set of equations to describe a complex waveform may lead to incorrect 

 results. 



3. Distantly Generated Tsunamis . 



When a tsunami travels a long distance across the ocean, the sphe- 

 ricity of the Earth must be considered to determine the effects of the 



47 



