results showed general agreement with the theoretical expressions for 

 the beat length and the second harmonic amplitude. 



The present data support these analyses to some extent. Secondary- 

 crests are apparent when S is greater than about 10, although the 

 critical value evidently depends slightly on d/L, for 0.05 < d/L < 0,13. 

 When a secondary crest is located halfway between the primary crests, the 

 primary crest is higher and sharper, confirming that the primary and 

 second harmonic waves are in phase there. However, the waveforms for 

 various H at the same (L,d) combination (Figs. B-3, B-4, and B-5) fail 

 to indicate a definite dependence of B/L on S. The waveform exhibits 

 a more complicated fine structure as H increases, but there is no 

 change in location of the secondary features and thus no change in the 

 beat length. Table B-5 gives the beat length between locations of con- 

 structive interference shown in the three steep propagating waveforms 

 presented in Figure B-2. The estimated beat length is approximately 

 3L/2, three times the prediction of Madsen (1971); however, these esti- 

 mates are somewhat qualitative and pertain to situations more highly 

 nonlinear than the stated range of validity of Madsen 's analysis. There 

 may be a slight decrease in beat length with increasing S, as Mei and 

 Unluata (1972) predicted. 



Table B-5. Beat lengths between primary and second 

 harmonics for three steep test waves in 

 96-foot tank (estimated from Fig, B-2), 



Wave period, 

 T 

 (s) 



Dimensionless 



water depth, 



d/L 



Dimensionless 

 Stokes parameter, 

 HL2/2d3 



Dimensionless 



beat length, 



B/L 



1.55 

 2,32 

 3,10 



0,124 

 0.079 

 0.057 



14 

 35 

 45 



1.7 

 1.5 

 1.4 



Several other nonlinear wave effects were observed in the 96-foot 

 tank tests, although no quantitative data are available. After a steep 

 crest passed a test pile, capillary waves were often radiated outward 

 from the front half of the pile. Theoretical treatments by Longuet- 

 Higgins (1963) and by Crapper (1970) have shown these waves can be gener- 

 ated where surface tension forces are accentuated due to a sharply curved 

 free surface. The blockage of the propagating waveform at the pile causes 

 an increased surface curvature. The resulting capillary waves are visible 

 in some photos in Section II. 



There was also some flow along steep crests in the 96-foot tank. A 

 slight lump on the crest was observed to bounce from one side of the tank 

 to the other as the crest propagated toward the beach. This lateral flow 

 was apparently neither the transverse wave described by Madsen (1974) nor 



89 



