II. WAVELENGTH MODIFICATION AND THE STOPPING OF WAVES 



When a current is absent, Peregrine (1976) l and the Shore Protection Man- 

 ual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 

 1977 ) 2 indicate that the wavelength, L, of a monochromatic wave is given by 

 L = La, where L A is computed by solving the "dispersion relation" 



2iTd T \ 



r (1) 



and 



J A 



tanh = the hyperbolic tangent 



T = the wave period 



g = the acceleration of gravity 32.2 feet (9.8 meters) per second 

 squared 



cL = the time-averaged (over enough wave periods to filter out waves) 

 water depth 



tt = the factor 3.14159... 



When a current is present, Peregrine (1976) 3 indicates that L is no 

 longer equal to L A but to L v , where L y is given by solving the equation 



[(1/T) - (V cos 6/L v )] 2 l /2TTd T \ 



2tt = — tanh (2) 



§ L V \ L V / 



where cos is the cosine, V the horizontal current speed, and the angle 

 between the horizontal current vector, _V, and the horizonal wave vector, JC. 

 The angle, 6, is taken to be greater than or equal to zero and less than or 

 equal to 180° (it radians). The vector _K has a magnitude of 2tt/L v and 

 points along the wave orthogonal in the direction of wave crest and trough 

 propagation. The relationship between horizontal current velocity, _V, hori- 

 zontal wave vector, K, wave orthogonal, wave crest, wave ray, and angle, 9, 

 is shown in Figure 1. The figure shows that when a current is present, wave 

 rays are not parallel to wave orthogonals and hence not perpendicular to wave 

 crests. 



Figure 2 shows a set of curves which represents the dimensionless solution 

 of equation (2). The F axis represents various values of the effective 

 Froude number 



1 PEREGRINE, D.H., "Interaction of Water Waves and Currents," Advances in 

 Applied Mechanics, Vol. 16, Academic Press, New York, 1976, pp. 9-117. 



2 U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shove 

 Protection Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1, 

 U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp. 



3 PEREGRINE, D.H. , op. cit. 



L0 



