In applying assumption (g) to waves in water of varying depth encoun- 

 tered when waves approach a beach the local depth is usually used. This 

 can be rigorously justified, but not without difficulty, for most practical 

 cases in which the bottom slope is flatter than about 1 on 10. A progres- 

 sive wave moving into shallow water will change its shape significantly. 

 Effects due to viscosity and vertical velocity on a permeable bottom may 

 be measurable in some situations, but these effects can be neglected in 

 most engineering problems. 



2.23 ELEMENTARY PROGRESSIVE WAVE THEORY (Small -Amplitude Wave Theory) 



The most fundamental description of a simple sinusoidal oscillatory 

 wave is by its length L (the horizontal distance between corresponding 

 points on two successive waves) ; height H (the vertical distance to its 

 crest from the preceding trough) ; period T (the time for two successive 

 crests to pass a given point) ; and depth d (the distance from the bed 

 to the Stillwater level). (See Appendix B for a list of common symbols.) 



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

 in the positive x-direction. The symbols used here are presented in the 

 figure. The symbol n denotes the displacement of the water surface 

 relative to the Stillwater level (SWL) and is a function of x and time. 

 At the wave crest, t\ 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 introduction of a velocity potential <t)(x, z, t) . Hori- 

 zontal and vertical components of the water particle velocities are defined 

 at a point (x, z) in the fluid as u = 8<(i/9x and w = 3(|)/3z. The velocity 

 potential, Laplace's equation, and Bernoulli's dynamic equation together 

 with the appropriate boundary conditions provide the necessary information 

 needed in deriving the small-amplitude wave formulas. Such a development 

 has been shown by Lamb (1932), Eagleson and Dean (See Ippen 1966b), and 

 others . 



2.231 Wave Celerity, Length and Period . The speed at which a wave form 

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

 distance traveled by a wave during one wave period is equal to one wave- 

 length, 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 



C = 



(2-2) 



2-7 



