water particle velocity at the wave crest becomes equal to the wave celerity. 

 This occurs when 



(x) 



= 0.78 (2-72a) 



max 



Laboratory investigations have shown that the value of (H/d) =0.78 

 agrees better with observations for oscillatory waves than for solitary waves . 

 Ippen and Kulin (1954), Galvin (1969), and Camfield and Street (1969) have 

 shown that the nearshore slope has a substantial effect on this ratio. Other 

 factors such as bottom roughness may also be involved. Galvin tested periodic 

 waves with periods from 1 to 6 seconds on slopes of m = 0.0, 0.05, 0.10, and 

 0.20, and found that H^/d^j ratios were approximately equal to 0.83, 1.05, 

 1.19, and 1.32, respectively. Camfield and Street tested single solitary 

 waves on slopes from m = 0.01 to m = 0.20 and found an empirical relationship 

 between the slope and the breaker height-to-water depth ratio given by 



H, 



■^ = 0.75 + 25m - 112m2 + 3870m3 (2-72b) 



b 



It was found that waves did not break when the slope m was greater than 

 about 0.18. It was also noted that as the slope increased the breaking posi- 

 tion moved closer to the shoreline. This accounts for the large values of 

 H^j/d^ for large slopes; i.e., as d^ -»• 0. In general, it must be concluded 

 that for some conditions, equation (2-72) is unsatisfactory for predicting 

 breaking depth. Further discussion of the breaking of waves with experimental 

 results is in Section VI. 



8. Stream-Function Wave Theory. 



In recent years, numerical approximations to solutions of hydrodynamic 

 equations describing wave motion have been proposed and developed by Dean 

 (1965a, 1965b, 1967) and Monkmeyer (1970). The approach by Dean, termed a 

 symmetric, stream-function theory, is a nonlinear wave theory that is similar 

 to higher order Stokes' theories. Both are constructed of sums of sine or 

 cosine functions that satisfy the original differential equation (Laplace 

 equation). The theory, however, determines the coefficient of each higher 

 order term so that a best fit, in the least squares sense, is obtained to the 

 theoretically posed, dynamic, free-surface boundary condition. Assumptions 

 made in the theory are identical to those made in the development of the 

 higher order Stokes' solutions. Consequently, some of the same limitations 

 are inherent in the stream-function theory; however, it represents a better 

 solution to the equations used to approximate the wave phenomena. It is more 

 important that the stream-function representation appears to more accurately 

 predict the wave phenomena observed in laboratory wave studies (Dean and 

 Le Mehaute, 1970), and may possibly describe naturally occurring wave phenom- 

 ena better than other theories. 



The long, tedious computations involved in evaluating the terms of the 

 series expansions that make up the higher order stream-function solutions make 

 it desirable to use tabular or graphical presentations of the solutions. 

 These tables, their use, and their range of validity have been developed by 

 Dean (1974). 



2-59 



