Shallow Water Wave Transformation through External Factors 125 



waves and a stronger sloping beach, this ratio h: 2 A can increase to 2-71. 

 The crest of the breaking wave is then at 0-65-0-85 x 2A above the surface 

 at rest. The approaching wave gives the impression of having a larger water 

 volume than the retreating wave. This may partly be due to an optical 

 illusion because, in accordance with the nature of the eddy, the water starts 

 already flowing back in the rear part of the wave, while in the front and 

 upper part the water is still advancing. According to Wheeler (1903, p. 37), 

 a substantial amount of water oozes through the gravel beach. 



Krummel (1911, vol. II, p. 110) emphasizes that the breaking of the 

 waves is not only dependent on the ratio between depth and wave height, 

 but also on the dimensions of the orbits in the deeper layers and at the 

 bottom. The breaking of the waves is, therefore, also influenced by a dis- 

 turbance of the horizontal water motion in greater depth. Gaillard found 

 that if the beach is shaped in terraces and its slopes vary between 1 : 30 and 

 1:90 the critical depth increases to 1-84 of the wave height. Coastal banks 

 which are located far into relatively deep water, or shoals in the open ocean 

 are generally visible by an increase in the swell. Often-times it produces 

 a heavy surf and thus the position of the shoals is indicated. According to 

 a compilation made by Krummel, surf phenomena occur for ocean depths 

 ranging from 15—20 to 200 m and over, probably as a consequence of 

 a terrace-like topography of the ocean bottom (Cialdi, 1860). These strong 

 currents extending into great depths without a particularly high visible swell 

 at the surface are called groundswell by seamen. 



More accurate observations and analyses have provided a theory about 

 the cause of a wave to break when approaching the shore. Sverdrup and 

 Munk (1946. p. 828) have elaborated on an interesting observation that 

 at the approximate moment of breaking, a wave behaves like the solitary 

 wave of Scott Russell. This seems to follow from the theory of shallow water 

 waves and of the solitary wave. According to the theory of Korteweg 

 and de Vries (1895) the profiles of the cnoidal waves correspond very 

 closely to those of waves in shallow water, just before they break. It was 

 shown on pp. 110 and 119 that the solitary wave is theoretically a special 

 case of these cnoidal waves. Therefore, it can be expected that the shallow 

 water waves behave in breaking like the solitary waves. 



When the water becomes so shallow that the ratio of depth h to wave 

 height H reaches the value of 1-28 (see p. 117), then the water at the crest 

 moves faster than the crest itself, and the wave breaks (Sverdrup and Munk). 

 From the theory of the solitary wave and from the assumption of conservation 

 of wave energy one can derive a relationship in which the ratio of wave 

 height H at the moment of breaking to wave height over deep water H 

 is a function of the initial steepness (ratio: wave height to wave length 

 in deep water) H /X . The solid line in Fig. 57 shows graphically this ratio. 

 The dots are observed values on the Atlantic and Pacific beaches. Although 



