given point), and depth d (the distance from the bed to the Stillwater 

 level, SWL) . (See App. B for a list of common symbols.) 



Figure 2-2 shows a two-dimensional, simple progressive wave propagating in 

 the positive x-direction, using the symbols presented above. The symbol n 

 denotes the displacement of the water surface relative to the SWL and is a 

 function of x and time t. At the wave crest, n is equal to the amplitude 

 of the wave a or one-half of the wave height. 



Small-amplitude wave theory and some finite-amplitude wave theories can be 

 developed by the introduction of a velocity potential (})(x, z, t) . Horizontal 

 and vertical components of the water particle velocities are defined at a 

 point (x, z) in the fluid as u = 8(t)/3x and w = 3(t)/9z. The velocity poten- 

 tial, Laplace's equation, and Bernoulli's dynamic equation together with the 

 appropriate boundary conditions, provide the necessary information to derive 

 the small-amplitude wave formulas. Such a development has been shown by Lamb 

 (1932), Eagleson and Eean (1966, see Ippen, 1966b), and others. 



a. Wave Celerity, Length, and Period . The speed at which a waveform 

 propagates is termed the phase velocity or wave celerity C. Since the dis- 

 tance traveled by a wave during one wave period is equal to one wavelength, 

 the wave celerity can be related to the wave period and length by 



C = ^ (2-1) 



An expression relating the wave celerity to the wavelength and water depth is 

 given by 



From equation (2-1), it is seen that equation (2-2) can be written as 



C = -p tanh l^^l (2-3) 



The values 2Tr/L and 2ti/T are called the wave number k and the wave 



angular frequency to, respectively. From equations (2-1) and (2-3) an 



expression for wavelength as a function of depth and wave period may be 

 obtained. 



Use of equation (2-4a) involves some difficulty since the unknown L appears 

 on both sides of the equation. Tabulated values of d/L and d/L (d/L« is 

 the deepwater wavelength) in Tables C-1 and C-2 in Appendix C may be used to 

 simplify the solution of equation (2-4a). Eckart (1952) gives an approximate 

 expression for equation (2-4a), which is correct to within about 5 percent. 

 This expression is given by 



gT2 ) /4it2 d\ 



2-7 



