and 



L = 



2lT 



L = - — = 5.12T2 ft (2-8b) 



If equations (2-7a) and (2-7b) are used to compute wave celerity when the rel- 

 ative depth is d/L = 0.25, the resulting error will be about 9 percent. It is 

 evident that a relative depth of 0.5 is a satisfactory boundary separating 

 deepwater waves from waves in water of transitional depth. If a wave is trav- 

 eling in transitional depths, equatior>.s (2-2) and (2-3) must be used without 

 simplification. Care should be taken to use equations (2-2) and (2-3) when 

 necessary; i.e., when the relative depth is between one-half and one-twenty- 

 fifth. 



When the relative water depth becomes shallow, i.e., 2Trd/L < 1/4 or d/L < 

 1/25, equation (2-2) can be simplified to 



= Vgd~ 



(2-9) 



This relation, attributed to Lagrange, is of importance when dealing with 

 long-period waves, often referred to as long waves. Thus, when a wave travels 

 in shallow water, wave celerity depends only on water depth. 



b. The Sinusoidal Wave Profile . The equation describing the free surface 

 as a function of time t and horizontal distance x for a simple sinusoidal 

 wave can be shown to be 



, = a cos ^-^ - -^j = - cos ^— - — j (2-10) 



where n is the elevation of the water surface relative to the SWL, and H/2 

 is one-half the wave height equal to the wave amplitude a. This expression 

 represents a periodic, sinusoidal, progressive wave traveling in the positive 

 x-direction. For a wave moving in the negative x-direction, the minus sign 

 before 2irt/T is replaced with a plus sign. When (2irx/L - 2irt/T) equals 0, 

 it/ 2, IT, 3ir/2, the corresponding values of n are H/2, 0, -H/2, and 0, 

 respectively. 



c. Some Useful Functions . It can be shown by dividing equation (2-3) by 

 equation (2-6), and equation (2-4) by equation (2-8) that 



I : : -:- : (2-n) 



o 

 If both sides of equation (2-11) are multiplied by d/L, it becomes 



(2-12) 



The term d/L has been tabulated by Wiegel (1954) as a function of d/L and 

 is presented in Appendix C, Table C-1. Table C-2 includes d/L as a function 



2-10 



