2.254 Subsurface Pressure . The pressure at any distance below the fluid 

 surface is given by 



H cosh [27r(z + d)/L] 

 ^§ 2 cosh (27rd/L) ^°' 



3 ttH^ tanh(27rd/L) 



8 



L sinhM27rd/L) 



cosh [47r(z + d)/L] 



sinh^ (27rd/L) 



cos r^ - ^ (2-56) 



ttH^ tanh (27rd/L) 



s'^T 



sinh^ (27rd/L) 



cosh 



4;r(z + d) 



2.255 Maximum Steepness of Progressive Waves . A progressive gravity wave 

 is physically limited in height by depth and wavelength. The upper limit, 

 or breaking wave height in deep water is a function of the wavelength, and 

 in shallow and transitional water is a function of both depth and wavelength. 



Stokes (1880) predicted theoretically that a wave would remain stable 

 only if the water particle velocity at the crest was less than the wave 

 celerity or phase velocity. If the wave height were to become so large 

 that the water particle velocity at the crest exceeded the wave celerity, 

 the wave would become unstable and break. Stokes found that a wave having 

 a crest angle less than 120 degrees would break (angle between two lines 

 tangent to the surface profile at the wave crest). The possibility of 

 existence of a wave having a crest angle equal to 120 degrees was shown by 

 Wilton (1914) . Michell (1893) found that in deep water the theoretical 

 limit for wave steepness was 



'H 



— 1 = 0.142 « - 

 "olmax ' 



(2-57) 



Havelock (1918) confirmed Michell 's findings. 



Miche (1944) gives the limiting steepness for waves traveling in 

 depths less than L^/2 without a change in form as 



27rd\ , /27rd\ 

 = 0.142 tanh 



(2-58) 



Laboratory measurements by Danel (1952) indicate that the above 

 equation is in close agreement with an envelope curve to laboratory 

 observations. Additional discussion of breaking waves in deep and 

 shoaling water is presented in Section 2.6, BREAKING WAVES. 



2.256 Comparison of the First- and Second-Order Theories . A comparison 

 of first- and second-order theories is useful to obtain insight about the 



2-39 



