e. 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° would break (angle between two lines tangent to the surface profile 

 at the wave crest) . The possibility of the existence of a wave having a crest 

 angle equal to 120° was shown by Wilton (1914). Michell (1893) found that in 

 deep water the theoretical limit for wave steepness was 



U^ = 0.142* j (2-57) 



\ o I max 



Havelock (1918) confirmed Michell 's findings. 



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

 less than Lo/2 without a change in form as 



Imax 



= 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 VI. 



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

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

 of a theory for a particular problem. It should be kept in mind that linear 

 (or first-order) theory applies to a wave that is symmetrical about the SWL 

 and has water particles that move in closed orbits. On the other hand, 

 Stokes' second-order theory predicts a waveform that is unsjonmetrical about 

 the SWL but still symmetrical about a vertical line through the crest and has 

 water particle orbits that are open. 



*************** EXAMPLE PROBLEM 7*************** 



GIVEN ; A wave traveling in water depth d = 6 meters (19.7 feet), with a wave- 

 length L = 60 meters (196.9 feet) and a height H = 1 meter (3.28 feet). 



FIND: 



(a) Compare the wave profiles given by the first- and second-order 

 theories. 



(b) What is the difference between the first- and second-order horizontal 

 velocities at the surface under both the crest and trough? 



2-37 



