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 analyti- 

 cally that Format 's principle and hence Snell's law followed from the 

 governing hydrodynamic equations, 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 pro- 

 cedures 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 constant. 

 (Orthogonals are lines drawn perpendicular to the wave crests, and extend 



in the direction of wave advance.) (See Figure 2-17.) 



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

 that is, in the direction of the orthogonals. 



(3) Speed of a wave of 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. 



2.32 GENERAL - REFRACTION BY BATHmETRY 



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

 function in the formula 



is nearly equal to unity, and Equation 2-2 reduces to 



" 2-n 



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

 in those regions deeper than one-half the wavelength (deep water) , refrac- 

 tion by bathymetry will not be significant. Where the water depth is 

 between 1/2 and 1/25 the wavelength (transitional water) , and in the region 

 where the water depth is less than 1/25 the wavelength (shallow water), 

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



2-65 



