(8) Determine U = -3— x -. • -r— 



b \ o b 



l\ 2 



(9) Find m and K(m) satisfying equation (10) using 



polynomial approximation of K given in Abramovitz 

 and Stegun (1964) 



(10) Determine E, A, B, f and then K from equation 

 (12) ^ 



(11) Go back to step (4) until K remains constant 



(12) Compare the obtained K value with value computed 

 from equation (26). If different, go back to step 

 (3) 



3. Results. 



The wave breaking angle, a^, as computed with linear, cnoidal- 

 linear, and cnoidal third-order Stokes approaches, is compared with the 

 experimental data of Kamphuis (1963) as shown in Le Mehaute and Koh (1967) 

 and Figure 12. These are apparently the only data on aj~, and are ob- 

 tained for a single bottom slope, S = 0.1. For Hq/Lq = 0.0175, linear 

 theory gives the best fit to the data. On the other hand, cnoidal 

 linear theory provides a better fit for the larger steepnesses, Hq/Lq = 

 0.053 and 0.062. Unfortunately, it is not possible to draw any de- 

 finite conclusions from the limited data. However, it appears that 

 linear theory provides the best estimate of wavelength irrespective 

 of wave steepness, as found by Eagleson and Dean (Ippen, 1966) . Also, 

 for large wave steepness, cnoidal theory predicts the wave height quite 

 accurately up to the point of breaking. 



For a given beach slope the waves break at increasing relative 

 depth ratios, d/L, as the deepwater steepness increases. For large 

 enough Hq/Lq, the cnoidal theory is no longer valid since it predicts 

 a nonphysical complex wave height (Svendsen, 1974) . The critical 

 deepwater wave steepness for which the cnoidal theory ceases to be 

 valid has been determined for five different bottom slopes (Table 2). 

 This critical value is only weakly sensitive to the magnitude of ag . 

 For Hq/Lq greater than the critical value, a different wave theory 

 such as Dean's (1974) or third-order Stokes (Le Mehaute and Koh, 1967) 

 must be used. 



In Figures 13 to 17, the variation of breaking wave angles 

 versus deepwater wave angle with Uq/Lo and S as parameters is de- 

 picted as computed using the cnoidal-linear theory. Similar results 

 for linear theory are shown in Figure 18, when H^/Lq and S are com- 

 bined into a single parameter. 



30 



