Brennen and Whitney 



-^(f)/[^-'] 



and i Is the a difference between the vertical walls. For T of 

 the order of 2 or 3 and for values of M up to . 3 , at least, there 

 was virtually no difference between the numerical and linearized 

 solutions. Figures 10 and 11 in which M = 0,344 demonstrate this. 



D, Example Three, Figs. 4(c), 13, 14, 15. A Sloping Beach 



By altering the condition on the boundary AB of example one 

 and employing the shoreline treatment of Section 5E, the interaction 

 of the waves with a sloping beach could be studied. In Fig, 13 a 

 small wave appraches a 27° beach. As the horizontal inclination of 

 the tangent to the free surface at the shoreline (p) decreases, the 

 shoreline (A) accelerates up the beach until P becomes positive. 

 The acceleration then reverses (as in Eq, (28)) and the wave reaches 

 maximum run up. The backwash is extremely rapid and positions 

 t/r = 21, 22 suggest that this causes the small wave which is follow- 

 ing the main one to break. By this time the cells have become very 

 distorted and the mesh points excessively widely spaced to allow 

 further progress. A similar succession of events takes place with 

 the larger wave and smaller beach angle (18°) of Fig. 14. Note in 

 this case the large run-up to wave-height ratio. In neither of these 

 cases does there appear to be any tendency for the main wave to 

 break on its approach run. Indeed the reaction with the beach is 

 similar to the behavior predicted by Carrier and Greenspan [ 1958] 

 in their non-linear shallow water wave analysis. The wave amplitude 

 was further Increased and the beach slope decreased to 9° In an 

 attempt to produce breaking on the approach run. A preliminary 

 result Is shown In Fig. 15. Variations In the application of the free 

 surface condition and In the shoreline treatment have, as yet, failed 

 to remove the Irregularities In that solution. A stronger shoreline 

 singularity coupled with an Insufficiently rigorous treatment of It may 

 be to blame. An optimistic viewer might detect a breaking tendency. 



E. Example Four, Figs. 4(d), 16. A Shelf 



One final example Is shown In Figs, 4(d) and 16 where the wave 

 travels up a shelf, created by changing the boundary condition on CD, 

 Fig. 1, Excessive vertical elongation of the cells on top of the shelf 

 caused this computation to be stopped at the last time shown, (At 

 this point the wave height/water depth ratio on the shelf Is of the order 

 of 2.) However, one can detect a splitting of the wave Into two waves 

 as might be expected from the theory of Lax [ 1968] , 



140 



