P = PgCXg ~ ^^ (2-63) 



i.e., 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 to 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^ = 1 - 10"^ 

 = 1 - 0.1 = 0.90). 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 to 2-15. 



The following problem illustrates the use of these figures. 



*************** EXAMPLE PROBLEM 8*************** 



GIVEN ; A wave traveling in water depth d = 3 meters (9.84 ft), with a 

 period T = 15 seconds, and a height H = 1 .0 meter (3.3 ft). 



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 



and 



M=0.33 



V? = 



T ,/f = 15 J^-i^ = 27.11 

 \ d 



From Figure 2-11, entering H/d and T Vg/d, determine the square of the 

 modulus of the complete elliptical integrals k . 



2-46 



