122 Shallow Water Wave Transformation through External Factors 



confirm or deny the small initial decrease in wave height. The observations 

 shown in Fig. 55 refer only to waves before they break during a short 

 interval immediately preceding the breaking, an increase in wave height 

 takes place, which is more rapid than the one given by the Stokes theory. 

 The reason for this descrepancy can be found in the rapid change in the 

 wave profile where the wave height is not any longer proportional to the 

 square root of the energy" (see p. 130). 



One of the most characteristic features of the onrushing swell on a sloping 

 beach in the form of breakers is the very regular formation of long-crested 

 waves. Jeffreys (1924) explains that, when a wave system composed of 

 different wave lengths advances into shallow water, the dissipation of energy 

 destroys the short-crested waves before the low crested waves of the swell. 



Airy (see Lamb, 1932, para. 188, p. 281) has shown that when the 

 elevation rj is not small compared with the mean depth h waves are no 

 longer propagated without change of type. The waves become steeper and 

 steeper because on the front slope of the wave the water particles are moving 

 faster, whereas on the rear slope of the wave the water particles move slower, 

 and we get to a point where the vertical accelerations can no longer be 

 neglected in comparison to the horizontal ones. The observation shows that 

 the wave topples over and breaks. Applying the method of successive ap- 

 proximations, we can derive the gradual development of an asymmetric wave 

 profile. The second approximation adds a second term to the simple wave 

 train which increases the duration of the fall and decreases the rising time 

 of the water surface. The wave profile has then the following form: 



r\ = acosx(x — ct)-\- . ^—^- xsm2>t{x— ct) . (V.21) 



The second term in the equation contains the factor x in the amplitude. 

 This means that this first approximation is valid only for a small distance x. 

 At a certain distance x the unsymmetry of the wave profile becomes so large 

 that the approximation (V.21) can hardly be maintained. This limit is at- 

 tained when the amplitude of the second term becomes equal to that of the 

 first one. This happens when 



5 f^ r - 1 



4 c 2 " x ~ l • 



The distance the wave can travel without overturning will be 



4 ^_ 

 3 gxa ' 



Fjelstad (1941) solved mathematically the differential equations for this 

 change of profile. He neglected only the vertical accelerations so that he was 

 justified to substitute for the pressure its hydrostatic value. He considered com- 



