35 



in equation 44 is to be obtained by differentiation with depth coiistant. 

 If these data are represented graphically with C a function of L for 

 different constant depths as parameter, tangents drawn to these curves 

 will intercept the axis of (21) at the group velocity. 



Section 6. WAVES ON A SLOPING BOTTOM 



None of the theories previously discussed applies to waves moving 

 over a sloping bottom. W. Wien (22) has considered the passage of a 

 deep-water wave over a discontinuity in the bottom. Rayleigh (23) 

 has discussed the variation in wave height on the assumption that 

 waves in each depth on a sloping bottom have the same characteristics 

 as in water of the same constant depth. 



The changes in height and length which waves experience on running 

 into shallow water is a well-known phenomenon. Quantitative knowl- 

 edge of this effect is important in field studies. The depth in which 

 the observations are made must be known in order to compute back 

 to the characteristics of the same waves in deep water. The same 

 problem arises in wave tank work where the most convenient method 

 of characterizing the waves used in an experiment is by means of the 

 computed characteristics in deep water. 



Airy (24) makes an interesting statement regarding oscillatory 

 waves over a shoaling bottom. 



It would appear that when the depth is variable, it is impossible that there 

 can be a series of waves which consist of oscillatory motion of the particles, and 

 which satisfy the equations of continuity and equal pressure. The following 

 physical interpretation of the mathematical result appears to be correct, and is 

 worthy of attention. It appears that, if the water is moving in the manner of 

 waves, one at least of the two conditions (continuity and equal pressure) must fail. 

 While the continuity holds, the equal pressure will exist from the nature of the 

 fluid. Therefore, the continuity must cease, or the water must become broken. 

 This appears to be the explanation of the broken water which is usually seen 

 upon the edge of a shoal or a ledge of rocks, although the whole is covered, per- 

 haps deeply, by the water. 



While Airy's conclusions appear to be substantiated by observation 

 of waves moving over a deeply submerged reef, waves traveling on a 

 gradually shoahng bottom do appear to move without breaking or 

 losing their identity. 



Stokes (2) commented on this problem as follows: 



When swells are propagated toward a smooth, very gently shelving shore, the 

 height increases where the finiteness of depth begins to take effect. * * * 

 The breaking is no doubt influenced by friction against the bottom, but I do not 

 beheve that it is wholly or even mainly due to this cause. Before the wave 

 breaks altogether the top gets very thin, but the maximum height for uniform 

 propagation is probably already passed by a good deal, so that we must guard 

 against being misled by this observation as to the character of the limiting form. 



A surprising conclusion reached by Ivanov (25) is that the wave 

 period depends upon water depth and bottom slope. For example, 



