CAUSES OF SURF 107 



the wave had offshore. The wave regains its initial height when it 

 reaches the point where the depth is about 0.06 as great as its own 

 initial length, after which it becomes higher than the height offshore 

 (fig.21). 34 



It is the initial length of the wave and not its height that determines 

 at what depth, i. e., at what point over the bottom slope, its shape be- 

 gins to change. A relatively long wave suffers greater deformation 

 than a relatively short one, because the alteration continues through 

 a greater range of depth in the first case than in the second. Conse- 

 quently, it is the initial steepness of the wave (i. e., the ratio between 

 its length and its height while still out in deep water) that determines 

 whether the increase in height that follows the initial decrease will 

 more than counterbalance the latter or not and, hence, whether the 

 wave will be appreciably higher at the time it breaks than it was 

 originally. Under most conditions on open coasts, breakers are high- 

 er than the waves. 



The following example may help to make this clear. Assume two 

 series of waves, both of them 2 feet high out in deep water, but one 

 series 500 feet long, the other only 100 feet. Measurable deformation 

 of the longer waves would commence when they reached a point 

 where the water was still about 250 feet deep. As they advanced 

 shoreward, their heights (after the slight initial decrease) would 

 theoretically increase to 2.1 feet by the time they reached the 20-foot 

 belt, to 2.7 feet by the time they reached a point where the water was 

 about 7 feet deep, and to 3.5 feet at the 5.5-foot line, where a wave of 

 this initial steepness might be expected to break. The shorter waves, 

 however, would continue their progress unaltered until they reached 

 a point where the depth was 40 feet, would decrease to a height of only 

 about 1.8 feet by the time they reached the 15-foot line, would then 

 build up again to the original height of 2 feet by the time they reached 

 the 5-foot line, and to 2.3 feet at the 3-foot line, where they might be 

 expected to break. 



34 The change in wave height is brought about by two opposing actions which may be 



represented in an equation //— h /' [ - 1 where H is the height at the selected point 



V 2 n C/Co 

 in shallow water, H is the height in deep water, C and C are corresponding values of 

 wave velocity, and n is a complicated function of depth and wave length : 



"(4) 



1 + - 



sinh 



(*r) 



where d is the depth of water, and L is the wave length in deep water. The first action 

 tends to decrease the wave height by making the value of n increase. But meantime the 

 wave velocity and hence wave length is decreasing, which causes the crests to peak up. 

 As shown in the equation, when C/C decreases the relative wave height must increase. 

 The resulting effect of the two opposing actions, therefore, is to cause a small initial de- 

 crease in the height followed by a rapid increase, until the wave becomes unstable and 

 breaks. The initial decrease, first theoretically predicted (O'Brien and others. 1942. 

 Techn. rep. U. S. Beach Erosion Board. No. 2. pp. 35-37) has been confirmed by tank 

 experiments and also by observations at the Scripps and Woods Hole Institutions. 



