where K, = (2TTdg/L), Hq is wave height in deep water (d/L>0.5), and 



Lq = (gT^/2Tr) is wavelength in deep water. The dashed curve in Figure 1 



solves equation (5) . 



For wave steepness greater than 0.015, the difference between H and 

 H^ is less than ± 10 percent, according to linear wave theory. Ignoring 

 this shoaling wave height change, equation (5) takes the simpler form: 



? sinh2 E. tanh C = 205.6 (— V (6) 



Table 1 shows E, remains less than 2t:(0. 179)ss9/8, so Maclaurin expansions 

 for the transcendental functions in equation (6) yield the accurate alge- 

 braic approximation 



where the omitted terms are on the order of (0.002 B,^^) . The solid 

 straight line in Figure 1 shows the solution of equation (7) if only the 

 first term on the left-hand side is retained; the dotted curve shows the 

 solution of equation (7) where the other terms are appreciable; the solu- 

 tion of equation (6) lies between the dotted and solid curves. According 

 to this development, the maximum water depth for intense bed agitation 

 may be obtained for a certain wave condition by reading the corresponding 

 (2TTdg/L) from Figure 1, and multiplying by {(L^/2Tr) tanh (2Trd3/L) } to 

 obtain dg . 



This development has considered the ideal two-dimensional situation 

 of a monochromatic wave shoaling according to linear wave theory. Grace 

 and Rocheleau (1973) concluded that available field measurements indicate 

 linear wave theory " ... provides an excellent prediction of the sample 

 mean, near-bottom water velocity beneath the crest of long design-type 

 waves...." LeMehaute, Divoky, and Lin (1968) reported laboratory measure- 

 ments showing linear wave theory accurately gives Uj, in near-breaking 

 waves. Thus, linear wave theory is a convenient and accurate first 

 approximation in the problem considered. 



III. COMPARISON OF CALCULATED RESULTS WITH LABORATORY PROFILES 



The results from a laboratory test of profile development with a 

 constant wave condition incident on an initially plane slope of fine sand 

 are shown in Figure 2. Changes in profile shape become minor after a 

 large number of waves, and many "equilibrium profiles" have been published 

 for various test conditions (see App. B) . Frequently, these profiles 

 exhibit a long, slightly sloping terrace (as in Fig. 2) , caused by off- 

 shore deposition of sand eroded from the initial slope above a certain 

 depth . 



Raman and Earattupuzha (19 72) pointed out the existence of stable 

 points occurring in wave development of a sand bed profile. If the waves 



10 



