consistent with present knowledge and in a readily usable form. 



The general problem would require the determination of the 

 shoaling and refraction of a multidirectional wave spectrum from deep 

 water over a randomly varying bottom topography until breaking occurs. 

 Although such an analysis is possible, it is much too complicated and 

 would have to be dealt with on a case by case basis using either 

 manual or computer methods. 



A significant and useful simplification is achieved by assuming 

 the bottom to be a uniformly sloped plane. This allows bottom vari- 

 ations to be described by a single parameter, i.e., the bottom slope 

 S. The refraction process is then described globally by Snell's law. 

 This discussion deals only with monochromatic waves under the usual 

 assumption that the wave period remains constant, and that friction 

 and reflection are ignored. 



To obtain accurate predictions it is necessary to have a wave theory 

 which is applicable up to the point of breaking. A lack of knowledge 

 of the actual breaking process requires the use of an empirical 

 breaking criterion to determine the point of breaking. A study by 

 Le Mehaute and Koh (1967) evaluated the Stokes first-, third-, and 

 fifth-order theories and compared the Miche (1944) breaking criterion 

 with an empirically derived equation. One of these equations was 

 dei'ived by fitting a number of experimental data points covering the 

 range 0.02 < S < 0.2 and 0.002 < Hq/Lq < 0.09. This equation ex- 

 plicitly accounts for beach slope and is 



Pj^= 0.76 5^/^^) (8) 



o ^"o 



Since equation (8) is based on observed data it takes into account 

 nonlinear effects such as wave height peak-up just before breaking. 

 In applying this equation to waves arriving at an angle to the shore, 

 Le Mehaute and Koh (1967) corrected the bottom slope for the angle of 

 incidence; however, they neglected to replace the deepwater wave 

 height with its unrefracted value. 



Subsequently, a new and easier approach to compute cnoidal 

 waves was presented by Svendsen (1974) . Brink-Kjaer and Jonsson 

 (1973) showed that near breaking the water depth is usually so shallow 

 that cnoidal theory applies. Indeed, it has been found that wave 

 height is described well by cnoidal theory in the area close to and 

 before breaking. 



In a recent report, Ostendorf and Madsen (1979) propose to use 

 cnoidal and linear Stokes wave theories in their respective areas of 

 applicability. A transition between the two theories which assumes 

 continuous variation of energy flux and phase velocities is also 

 presented. Ostendorf and Madsen further suggest the use of an em- 

 pirical breaking criterion which is sensitive to bottom slope and 

 depth-varying wave parameters, i.e.. 



24 



