Thus, second-order theory predicts a pressure, 



p = 64,441 + 462 = 64,903 N/m^ (1,356 lbs/ft^) 



(e) Evaluation of the hydrostatic pressure component (60,270 newtons per 

 square meter) (1,288 pounds per square foot) indicates that Airy theory 

 gives a dynamic component of 4171 newtons per square meter (107 pounds 

 per square foot) while Stokes theory gives 4633 newtons per square 

 meter (121 pounds per square foot). Stokes theory shows a dynamic 

 pressure component about 11 percent greater than Airy theory. Using 

 equation (2-38) , the energy in one wavelength per unit width of crest 

 given by the first-order theory is 



^ . PgH^L . (1025)(9.8)(l)a(60) . ^^_^^^ ^^^ ^,^_,^„ ^^.^^^^^^, 

 8 8 



*************************************** 



6. Cnoidal Waves . 



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

 water are frequently best described by cnoidal wave theory. The existence in 

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

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

 Korteweg and DeVries (1895). The term ano-idal 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 pre- 

 sented 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 deter- 

 mined by Laitone (1963) and others is d/L < 1/8; and the Ursell or Stokes 

 parameter, is L^H/d^ > 26 (see Fig. 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 sig- 

 nificance, it is used to express the relationships between the various wave 



2-44 



