result in particle velocities at the wave crest greater than the wave 

 celerity and, consequently, instability. 



Limiting steepness -p- = 0.142 



Figure 2-64. Wave of Limiting Steepness in Deep Water 



2.62 ■ SHOALING WATER 



When a wave moves into shoaling water, the limiting steepness which 

 it can attain decreases, being a function of both the relative depth d/L, 

 and the beach slope m, perpendicular to the direction of wave advance. 

 A wave of given deepwater characteristics will move toward a shore until 

 the water becomes shallow enough to initiate breaking, this depth is 

 usually denoted as d]-, and termed the breaking depth. Munk (1949) derived 

 several relationships from a modified solitary wave theory relating the 

 breaker height H^, the breaking depth d^,, the unrefracted deepwater 

 wave height H^, and the deepwater wavelength L^. His expressions are 

 given by 





1 



3.3(HyL^)H 



(2-89) 



and 



H. 



= 1.28, 



(2-90) 



The ratio H^/H^ is frequently termed the breaker height index. Subse- 

 quent observations and investigations by Iversen (1952, 1953), Galvin 

 (1969), and Coda (1970) among others, have established that H2,/H^ and 

 dy/U-jj depend on beach slope and on incident wave steepness. Figure 2-65 

 shows Coda's empirically derived relationships between H^j/H^ and H^/L^ 

 for several beach slopes. Curves shown on the figure are fitted to widely 

 scattered data; however they illustrate a dependence of H^j/H^^ on the 

 beach slope. Empirical relationships between d^/H^, and H^^/gT^ for 

 various beach slopes are presented in Figure 2-66. It is recommended 

 that Figures 2-65 and 2-66 be used, rather than Equations 2-89 and 2-90, 

 for making estimates of the depth at breaking or the maximum breaker 



2-121 



