that is, the pressure distribution may be assumed to vary linearly from 

 pgyg at the bed to zero at the surface. 



Figures 2-9 and 2-10 show the dimensionless cnoidal wave surface 

 profiles for various values of the square of the modulus of the elliptic 

 integrals k^, while Figures 2-11 through 2-15 present dimensionless 

 plots of the parameters which characterize cnoidal waves. The ordinates 

 of Figures 2-11 and 2-12 should be read with care, since values of k^ 

 are extremely close to 1.0 (k^ = l-lO"^ = 1-0. 1 = 0.99) . It is the 

 exponent a of k^ = 1-10"" that varies along the vertical axis of 

 Figures 2-11 and 2-12. 



Ideally, shoaling computations might best be performed using cnoidal 

 wave theory since this theory best describes wave motion in relatively 

 shallow (or shoaling) water. Simple, completely satisfactory procedures 

 for applying cnoidal wave theory are not available. Although linear wave 

 theory is often used, cnoidal theory may be applied by using figures such 

 as 2-9 through 2-15. 



The following problem will illustrate the use of these figures. 



************** EXAMPLE PROBLEM ************** 



GIVEN : A wave traveling in water depth of d = 10 feet, with a period of 

 T = 15 seconds, and a height of H = 2.5 feet. 



FIND : 



(a) Using cnoidal wave theory, find the wavelength L and compare this 

 length with the length determined using Airy theory. 



(b) Determine the celerity C. Compare this celerity with the celerity 

 determined using Airy theory. 



(c) Determine the distance above the bottom of the wave crest (y^) 

 and wave trough (y^) . 



(d) Determine the wave profile. 

 SOLUTION: 



(a) Calculate 



H 2.5 



T = — = 0.25 



d 10 



and 



'' ^ "' Jw - ^'■'' 



2-49 



