Thus, second-order theory predicts a pressure, 



p = 1395 + 14 = 1409 lbs/ft^ . 



(e) Using Equation 2-38, the energy in one wavelength per unit width 

 of crest given by the first-order theory is: 



pgtfL (2) (32.2) (4)^200) ^^„^^ft-lbs 



E = ^ = ^^^-^^ i^jL_:^ .' = 25,800 —r- • 



8 8 ft 



Evaluation of the hydrostatic pressure component (1288 lbs/ft^) 

 indicates that Airy theory gives a dynamic component of 107 lbs/ft^ 

 while Stokes theory gives 121 lbs/ft^. Stokes theory shows a 

 dynamic pressure component about 13 percent greater than Airy 

 theory. 



2.26 CNOIDAL WAVES 



Long, finite-amplitude waves of permanent form propagating in shallow 

 water are frequently best described by anoidal wave theory. The existence 

 in shallow water of such long waves of permanent form may have first been 

 recognized by Boussinesq , (1877) . However, the theory was originally 

 developed by Korteweg and DeVries (1895) . The term anoidal is used since 

 the wave profile is given by the Jacobian elliptical cosine function 

 usually designated en . 



In recent years, cnoidal waves have been studied by many investigators. 

 Wiegel (1960) summarized much of the existing work on cnoidal waves, and 

 presented the principal results of Korteweg and DeVries (1895) and Keulegan 

 and Patterson (1940) in a more usable form. Masch and Wiegel (1961) 

 presented such wave characteristics as- length, celerity and period in 

 tabular and graphical form, to facilitate application of cnoidal theory. 



The approximate range of validity for the cnoidal wave theory as 

 determined by Laitone (1963) and others is d/L < 1/8, and the Ursell 

 parameter, L^H/d^ > 26. (See Figure 2-7.) As wavelength becomes long, 

 and approaches infinity, cnoidal wave theory reduces to the solitary wave 

 theory which is described in the next section. Also, as the ratio of wave 

 height to water depth becomes small (infinitesimal wave height), the wave 

 profile approaches the sinusoidal profile predicted by the linear theory. 



Description of local particle velocities, local particle accelerations, 

 wave energy, and wave power for cnoidal waves is difficult; hence their 

 description is not included here, but can be obtained in graphical form 

 from Wiegel (1960, 1964) and Masch (1964). 



Wave characteristics are described in parametric form in terms of the 

 modulus k of the elliptic integrals. While k itself has no physical 



2-4- 



