toward deep water. In such cases, a sheaf or fan of orthogonals may be 

 projected seaward in directions some 5 or 10 degrees apart. See Figure 

 2-22a. With the deepwater directions thus determined by the individual 

 orthogonals, companion orthogonals may be projected shoreward on either 

 side of the seaward projected ones to determine the refraction coefficient 

 for the various directions of wave approach. (See Figure 2 -22b.) 



2.325 Other Graphical Methods of Refraction Analysis . Another graphical 

 method for the construction of refraction diagrams is the wave-front 

 method (Johnson, et al., 1948). This method is particularly applicable 

 to very long waves where the crest alignment is also desired. The method 

 is not presented here, where many diagrams are required, because, where 

 many diagrams are required, it is overbalanced by the advantages of the 

 orthogonal method. The orthogonal method permits the direct construction 

 of orthogonals and determination of the refraction coefficient without the 

 intermediate step of first constructing successive wave crests. Thus, 

 when the wave crests are not required, significant time is saved by using 

 the orthogonal method. 



2.326 Computer Methods for Refraction Analysis . Harrison and Wilson (1964) 

 developed a method for the numerical calculation of wave refraction by use 

 of an electronic computer. Wilson (1966) extended the method so that, in 

 addition to the numerical calculation, the actual plotting of refraction 

 diagrams is accomplished automatically by use of a computer. Numerical 

 methods are a practical means of developing wave refraction diagrams when 



an extensive refraction study of an area is required, and when they can 

 be relied upon to give accurate results. However, the interpretation of 

 computer output requires care, and the limitations of the particular scheme 

 used should be considered in the evaluation of the results. For a dis- 

 cussion of some of these limitations, see Coudert and Raichlen (1970). 

 For additional references, the reader is referred to the works of Keller 

 (1958), Mehr (1962), Griswold (1963), Wilson (1966), Lewis, et al., (1967), 

 Dobson (1967), Hardy (1968), Chao (1970), and Keulegan and Harrison (1970), 

 in which a number of available computer programs for calculation of refrac- 

 tion diagrams are presented. Most of these programs are based on an algo- 

 rithm derived by Munk and Arthur (1951) and, as such, are fundamentally 

 based on the geometrical optics approximation. (Fermat's Principle.) 



2.327 Interpretation of Results and Diagram Limitations . Some general 

 observations of refraction phenomena are illustrated in Figures 2-23, 24, 

 and 25. These figures show the effects of several common bottom features 

 on passing waves. Figure 2-23 shows the effect of a straight beach with 

 parallel evenly spaced bottom contours on waves approaching from an angle. 

 Wave crests turn toward alignment with the bottom contours as the waves 

 approach shore. The refraction effects on waves normally incident on a 

 beach fronted by a submarine ridge or submarine depression are illustrated 

 in Figure 2-24a and 2-24b. The ridge tends to focus wave action toward 

 the section of beach where the ridge line meets the s horeli ne. The ortho- 

 gonals in this region are more closely spaced; hence-v/b^/o is greater 

 than 1.0 and the waves are higher than they would be if no refraction 

 occurred. Conversely, a submarine depression will cause orthogonals to 



2-75 



