when the water particle velocity at the wave crest becomes equal to the 

 wave celerity. This occurs when 



(2-72) 



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) and Galvin (1969) have shown that the near- 

 shore slope has a substantial effect on this ratio. Other factors such as 

 bottom roughness may also be involved. For slopes of 0.0, 0.05, 0.10, and 

 0.20, Galvin found that H/d ratios were approximately equal to 0.83, 

 1.05, 1.19, and 1.32, respectively. Thus, 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 2.6 - BREAKING WAVES. 



2.28 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 which is 

 similar to higher order Stokes' theories. Both are constructed of suras 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 con- 

 dition. 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. More important is that the stream function representation 

 appears to better predict some of the wave phenomena observed in laboratory 

 wave studies (Dean and L^Mehaut^, 1970), and may possibly describe naturally 

 occurring wave phenomena 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 range of validity have been 

 developed by Dean (1973). 



2.3 WAVE REFRACTION 



2.31 INTRODUCTION 



Equation 2-2 shows that wave celerity depends on the depth of water 

 in which the wave propagates. If the wave celerity decreases with depth, 

 wavelength must decrease proportionally. Variation in wave velocity occurs 



2-62 



