The decrease in wave celerity with decreasing water depth can be consid- 

 ered similar to the decrease in the speed of light with an increase in the 

 refractive index of the transmitting medium. Using this analogy, O'Brien 

 (1942) suggested the use of Snell's law of geometrical optics for solving the 

 problem of water-wave refraction by changes in depth. The validity of this 

 approach has been verified experimentally by Chien (1954), Ralls (1956), and 

 Wiegel and Arnold (1957). Chao (1970) showed analytically that Fermat's prin- 

 ciple and hence Snell's law followed from the governing hydrodynamic equa- 

 tions, and was a valid approximation when applied to the refraction problem. 

 Generally, two basic techniques of refraction analysis are available — 

 graphical and numerical. Several graphical procedures are available, but 

 fundamentally all methods of refraction analyses are based on Snell's law. 



The assumptions usually made are 



(1) Wave energy between wave rays or orthogonals remains con- 

 stant. (Orthogonals are lines drawn perpendicular to the wave crests, 

 and extend in the direction of wave advance.) (See Fig. 2-17.) 



(2) Direction of wave advance is perpendicular to the wave crest; 

 i.e., in the direction of the orthogonals. 



(3) Speed of a wave with a given period at a particular location 

 depends only on the depth at that location. 



(4) Changes in bottom topography are gradual. 



(5) Waves are long-crested, constant period, small-amplitude, and 

 monochromatic . 



(6) Effects of currents, winds, and reflections from beaches, and 

 underwater topographic variations are considered negligible. 



2. General — Refraction by Bathymetry . 



In water deeper than one-half the wavelength, the hyperbolic tangent 

 function in the formula 



, gL /2TTd 

 C2 = -2- tanh ' 

 2tt 



is nearly equal to unity, and equation (2-2) reduces to 



o 2Tr 



In this equation, the velocity C^ does not depend on depth; therefore, in 

 those regions deeper than one-half the wavelength (deep water) , refraction by 

 bathymetry will not be significant. Where the water depth is between one-half 

 and one-twenty-fifth the wavelength (transitional water), and in the region 

 where the water depth is less than one-twenty-fifth, the wavelength (shallow 

 water), refraction effects may be significant. In transitional water, wave 



2-62 



