642 GROEN AND GROVES [CHAP. 17 



obtained by simple reflection of the ordinary sinusoidal waves at sea. A more 

 complicated model, but one much more satisfying from the standpoint of 

 similarity with a real coast, is an infinitely long beach of constant slope. In this 

 case we deal with the half-plane x>0 with the boundary condition Ux{0, y, t) = 0, 

 and set h = sx. Let us consider again a flat ocean with waves of small amplitude 

 and use the linearized equations (3) and (4). 



Let us look for free edge-wave solutions of the form 



Ux = X{x) exp [i{ky + cTt)], 



Uy = Y{x) ex-p[i{ky + at)], 



C = Z{x)exip[i{ky + at)], 



considering k to be positive and a to be either positive or negative. Then the 

 functions X, Y, Z, must satisfy 



iaX-fY + gsxZ' = 



fX + iaY + igsxZ = (54) 



X' + ikY + iaZ = 



with -X'(O) = 0. The functions X and Y can be eliminated from (54) yielding 



where 



. = '-^^f± (56) 



gs a 



The only solutions of (55) giving finite Z for all x > are the functions 



Z = Aqn{2kx), (57) 



where q-n is the Laguerre function (see Reid, 1958, or Eckart, 1951) of order n. 

 The boundary condition is also satisfied by (57). But (57) is a solution only if 



K = {2n+\)k. (58) 



Equations (56) and (58) define a relationship between k and o- from which the 

 longshore phase velocity, c — a/k, and the group velocity, dajdk, can readily be 

 obtained. This relationship is cubic in a ; the three roots correspond to distinct 

 modes for each order n. The order of the Laguerre function determines the 

 onshore-offshore wavelength (which varies with distance from shore) ; the 

 higher the n the shorter the wavelength. Greenspan (1956) has shown that 

 storms with usual dimensions will excite the fundamental or zero order to a 

 greater extent than the higher orders. Reid (1958) has shown that for the zero 

 order only two modes are physically significant. The corresponding roots are 



oi= -U-Vigsk+ip) 

 ^2= -U+Vigsk+y^). 



In the case of a rotationless earth, /= and the two modes have frequencies 

 given by CT= ± \/{gsk). That is, the waves are propagated along the coast in 



