124 Shallow Water Wave Transformation through External Factors 



angle formed by the orbit and the vertical plane exceeds 45°, the transversal 

 horizontal component of motion in the ellipse will exceed the vertical one. 

 In addition to this change of plane of the orbits, there is a decrease in the 

 velocity of propagation of the wave, as a consequence of decreasing depth. 

 The wave crests turn parallel with the shore line (see p. 127). Bjerknes (1921) 

 has illustrated by two examples the geometrical viewpoint when the waves 

 run on to the shore (Fig. 56). In connection with these presentations, we 



Fig. 56. Wave advancing on a straight even beach. Horizontal projection of streamlines 

 (full lines) and orbits (dotted), a: westward; b: eastward travelling wave; shore sloping 



slightly to the north. 



should always remember that for the dynamics of the wave process, the 

 vertical displacements always remain decisive; for the exterior form of the 

 process, the horizontal components of motion are decisive. Case a deals 

 with a wave travelling westward in ocean towards a beach sloping up north- 

 ward; case b deals with a wave which travels eastward, the topography of 

 the bottom remaining the same as in case a. The orbits and the horizontal 

 component of the streamlines have been drawn in their projection on the 

 horizontal plane in the figure. In case a, the orbital motion at the shore is 

 anticyclonic and the waves run up the beach area from east to west ; in case 

 b, on the contrary, it is cyclonic and the waves run up from west to east. The 

 rolling of the ocean waves up the well delineated and gently rising beach 

 area is a regular process. 



The wave length of the waves rolling on to the shore becomes shorter 

 and, consequently, the wave height increases. With waves of great amplitude, 

 this increase causes them to topple over, and the wave process degenerates 

 into an eddy with a horizontal axis. Several authors have measured the 

 critical depth over which waves will start breaking (see Scott Russell, 

 1844, p. 352; Bazin, 1865, p. 509; Stevenson, 1864, p. 72; Gaillard, 

 1935; and Forchheimer, 1924, p. 379). It is assumed that the waves start 

 to break as soon as the wave height equals the water depth. From observations 

 in nature it appears, however, that with head-wind or in calm weather the 

 wave starts breaking when the depth beneath the undisturbed level is 0-72 

 of the wave height 2A. With a wind in the direction of propagation of the 



