which also applies for nonlinear as well as for linear theory. The 

 subscript o refers to deepwater wave characteristics. 



In many cases, the refraction method provides a reasonably 

 accurate measure of the changes waves undergo on approaching a coast. 

 However, if the angle of a wave ray with the bottom contour is large 

 (i.e., larger than 70°), minor error in the value of the incident 

 angle leads to a large error in direction angle a in shallow water. 

 Also, accuracy as far as height changes are concerned cannot be ex- 

 pected where bottom slopes are steeper than 1/10. No strict limit 

 has been set, but the accuracy of wave heights derived from orthogonals 

 that bend sharply is questionable. In short, refraction coefficients 

 which are quite different from unity, such as K^. < 0.5 and K.^ > 1.5, 

 must be doubted (Whalin, 1971] . 



Nonlinear effects, having an effect on wavelength, phase and 

 group velocity and energy flux, subsequently have an effect on wave 

 refraction. This problem has been examined by Chu (1975) who used a 

 mix of three theories, i.e., the first-order cnoidal theory of Korteweg 

 and DeVries (1895), the second-order hyperbolic wave of Iwagaki (1968), 

 and the Stokes third-order wave as given by Le Mehaute and Webb (1964), 

 which led to some inconsistencies in approximations. Skovgaard and 

 Petersen (1977) used instead the first-order cnoidal theory of Svendsen 

 (1974) and the stream function wave theories of Dean (1970) . 



Theoretically, it is possible to express phase velocities as a 

 function of the relative wave heights from nonlinear wave theories. 

 For example, the deepwater wavelength at a third-order Stokesian 

 approximation and the breaking wavelength by a cnoidal or hyperbolic 

 wave theory can be conveniently expressed. However, it is interesting 

 that due to deformation of wave profile on a sloped bottom, the simple 

 linear theory has been verified (experimentally) quite well (Ippen, 

 1966) . Wavelengths given by linear and cnoidal theories are compared 

 in Figure 10. Although the cnoidal theory predicts wave height well up 

 to breaking, it overpredicts wavelengths significantly. Cnoidal theory, 

 in fact, predicts an increase in wavelength for a decrease in depth 

 when the relative height, H/d, is sufficiently large. This increase 

 is not reflected by known data (Ippen, 1966) which are fitted quite 

 well by linear theory (Fig. 11). 



III. BREAKING WAVE CHARACTERISTICS ON A SLOPED 

 PLANE BEACH 



1 . Review of Previous Work , 



The determination of longshore currents and sediment transport 

 depends crucially on the characteristics of the breaking wave field. 

 The wave energy transport, or energy flux, is of particular importance 

 such that accurate determination of wave height, wavelength, depth at 

 breaking, and breaking wave angle becomes essential. 



This section deals with the practical aspects of determining 

 the breaking wave characteristics when certain deepwater character- 

 istics are given. The objective is to derive and present results 



22 



