Recently J. J» Stoker (14-) has extended the non-linear shallow 

 water wave theory by means of methods derived for the study of unsteady 

 flow in one dimension of a compressible gas. The theory is approximate, 

 and application is lacking, but it is interesting to note that the pro- 

 cedure permits the analysis of unsteady motions and can perdict the 

 wave form at all points up a beach to the breaker* The continuous wave 

 form so derived becomes assymetrical as the breaker line is approached, 

 with the wave front slope steeper than that behind the wave crest. All 

 other theories however, approximate the unsteady motion up a beach by 

 a series of different steady motions. The assumption is that at every 

 depth on a slope, the -wave will behave as if it were advancing over "a 

 horizontal bettom at that depth. The wave form then is predicted approx- 

 imately by a series of still pictures, instead of a continuous record. 

 Munk's theory in particular predicts a breaker which on the whole is 

 symmetrical in shape, while Stoker's development predicts a marked steep- 

 ening of the wave front and a very unsymmetrical shape for the waves at 

 breaking. (15) 



The theory most commonly used for the prediction of wave parameters 

 is that of progressive oscillatoiy waves of small amplitude. This theory 

 as with Stokes' second approximation for waves of finite amplitude gives 

 for the wave velocity 



(9) C2 = ZL ^^^^ £|V 



To obtain an expression for the change of wave height with depth, 

 the assumption is made that the wave form approximates a sine curve (1°) 

 (or better that the effect of higher order terms may be ignored). 

 That is 



(10) y= #^^^^-1^ 



The potential energy per unit surface area is given by 



(u) %>- ^' 



and the kinetic energy is numerically equal, therefore the total energy 

 is 



(12) H-"^' 



It has been shown for both deep water (18) and sha!:^low water waves (19) 

 that of this energy only a portion i^ transmitted forward with the wave 

 form, and that this portion is given by the ratio of group velocity to 



10 



