equations of motion. For shallow beaches, with typical slopes less than 0.1, 

 Eckart's solution yields very accurate results compared to Ursell's full solution, 

 especially for the lowest mode edge waves (Schaffer and Jonsson 1992). The 

 general mathematical description for edge waves propagating alongshore is 

 written as 



r\(x,y,t) = a <$>(x) cos(k y - cor) (1) 



where 



*1 



= elevation 



X 



= cross-shore coordinate, at shoreline, positive offshore 



y 



= longshore coordinate 



t 



= time 



a 



= edge wave shoreline amplitude 



4> 



= cross-shore amplitude function 



K 



= longshore wave number 



G) 



= radial frequency 



Several solutions satisfy the edge wave boundary conditions for a given 

 frequency and correspond to different edge wave modes. For Eckart's (195 1) 

 solution on a plane beach of slope (3, these waves satisfy the shallow-water 

 dispersion relation 



co 2 = gk [(2n + l) tanp] (2) 



where the mode number n = 0, 1, 2..., subject to the condition that 

 (2« + 1) tanP < 1 . Thus, for a specific frequency, mode edge waves have the 

 largest longshore wave numbers (smallest longshore wavelengths) with higher 

 modes having increasingly smaller wave numbers (larger longshore 

 wavelengths). The possible number of edge wave modes is bound by the 

 condition (2« + 1 ) tan P < 1 , with the possible number of modes increasing as 

 the beach slope decreases. This defines a cutoff for the highest possible mode 

 since the longshore wave number is limited by the deepwater wave number 

 (k = o> 2 /g). Thus, a discrete set of edge wave solutions exists for longshore 



wave numbers ky in the range co 2 /g < \k\ < <o 2 /gp\ If k <(o 2 /g, then the 



wave will enter deep water and not be refractively trapped to the shore. This is 

 the region for leaky wave propagation, with a continuum of solutions with 

 longshore wave numbers smaller than the deepwater cutoff (Figure 3). 



Edge and leaky waves have long wavelengths relative to their amplitudes, 

 allowing for near perfect reflection from the shoreline without breaking. These 

 waves set up a standing wave structure with nodes and anti-nodes in the cross- 

 shore, and can be either progressive or standing in the longshore. The cross- 

 shore amplitude function <j>(*) was found by Eckart (1951) for 



Chapter 2 Infragravity Wave Dynamics 



