subscript is omitted. (Ippen, 1966b, pp 21-240 If units of feet and 

 seconds are specified, the constant g/2Tr is equal to 5.12 ft/sec^ and 



and 



C = ^ = 5.12 T (ft/sec) , (2-7) 



" 2-n 



L^ = ^— = 5.12 T^ (ft). (2-8) 



If Equation 2-7 is used to compute wave celerity when the relative depth 

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

 dent 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 

 traveling in transitional depths. Equations 2-2 and 2-3 must be used with- 

 out simplification. Care should be exercised to use Equations 2-2 and 2-3 

 when necessary, that is, when the relative depth is between 1/2 and 1/25. 



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

 d/L < 1/25, Equation 2-2 can be simplified to 



C = yf^. (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. 



2.232 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 



17 = a cos I — 1 = — cos I — -— ) , (2-10) 



where n is the elevation of the water surface relative to Stillwater 

 level, 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, one need only replace the minus sign before 2TTt/T with a 

 plus sign. When (2Trx/L - 2Trt/T) equals 0, Tr/2, tt, 37v/2, the corresponding 

 values of n are H/2, 0, - H/2, and 0, respectively. 



2.233 Some Useful Functions . It can be shown by dividing Equation 2-3 by 

 Equation 2-6, and by dividing Equation 2-4 by Equation 2-8 that 



(2-11) 



2-10 



