seaward in directions some 5° or 10° apart (see Fig. 2-22, a). With the deep- 

 water 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 Fig. 2-22, b). 



e. Other Graphical Methods of Refraction Analysis . Another graphical 

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

 (Johnson, O'Brien, and Isaacs, 1948). This method applies particularly to 

 very long waves where the crest alinement is also desired. The method is not 

 explained here where many diagrams are required because it would be over- 

 balanced 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, signif- 

 icant time is saved by using the orthogonal method. 



f. 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 addi- 

 tion 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 discussion of some of 

 these limitations, see Coudert and Raichlen (1970). For additional refer- 

 ences, the reader is referred to the works of Keller (1958), Mehr (1962), 

 Griswold (1963), Wilson (1966), Lewis, Bleistein, and Ludwig , (1967), Itobson 

 (1967), Hardy (1968), Chao (1970), and Keulegan and Harrison (1970), in which 

 a number of available computer programs for calculation of refraction diagrams 

 are presented. Most of these programs are based on an algorithm derived by 

 Munk and Arthur (1951) and, as such, are fundamentally based on the geomet- 

 rical optics approximation (Fermat's principle). 



g. Interpretation of Results and Diagram Limitations . Some general 

 observations of refraction phenomena are in Figures 2-23, 2-24, and 2-25. 

 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 alinement with the bottom contours as the waves approach shore. The 

 refraction effects on waves normally incident on a beach fronted by a subma- 

 rine ridge or submarine depression are illustrated in Figure 2-24 (a and b) . 

 The ridge tends to focus wave action toward the section of beach where the 

 ridge line meets the s horeli ne. The orthogonals in this region are more 

 closely spaced; hence Vb^/b is greater than 1.0 and the waves are higher 

 than they would be if no refraction occurred. Conversely, a submarine depres- 

 sion will cause orthogonals to diverge, resulting in low heights at the shore 

 (b /b less than 1.0). Similarly, heights will be greater at a headland than 

 in a bay. Since the wave energy contained between two orthogonals is con- 

 stant, a larger part of the total energy expended at the shore is focused 

 on projections from the shoreline; consequently, refraction straightens an 



2-71 



