ti = r\'(x)e 



Vr^ ( V-"0 



(14) 



Substituting Equations 12-14 into the shallow-water Equations 9-11 results in 



aw = -g 



dx 



(15) 



av' = —u' - gkW 



dx * 



(16) 



en/ = — i '- - khv 



dx 



(17) 



where a = (k V - co) . These equations can be manipulated to obtain two 

 second-order differential equations in terms of n, and u 



gh\l &c\ 



a ) \ dx 



Kgh 







(18) 



d | hu 



(1 - y 2 ) dx\ a 



-Y 2 ^ ™\ =o 



(19) 



where 



Y 2 = 



k$gh 



(20) 



In the case of no alongshore currents ( V = ), Equation 1 8 can be written in the 

 form of Eckart's (1951) edge wave equation 



~L[ gh &) + n(&>2 + ghk2y) = ° (21) 



The effective beach profile is defined by Howd, Bowen, and Holman (1992) 



h'(x) 



h(x) 



1 - 



V(x) 



(22) 



20 



Chapter 2 Infragravity Wave Dynamics 



