irregular coast. Bottom topography can be inferred from refraction patterns 

 in aerial photography. The pattern in Figure 2-17 indicates the presence of a 

 submarine ridge. 



Refraction diagrams can provide a measure of changes in waves approaching 

 a shore. However, the accuracy of refraction diagrams is limited by the 

 validity of the theory of construction and the accuracy of depth data. The 

 orthogonal direction change (eq. 2-78a) is derived for straight parallel con- 

 tours. It is difficult to carry an orthogonal accurately into shore over 

 complex bottom features (Munk and Arthur, 1951). Moreover, the equation is 

 derived for small waves moving over mild slopes. 



Dean (1974) considered the combined effects of refraction and shoaling, 

 including nonlinearities applied to a slope with depth contours parallel to 

 the beach but not necessarily of constant slope. He found that nonlinear 

 effects can significantly increase (in comparison with linear theory) both 

 amplification and angular turning of waves of low steepness in deep water. 



Strict accuracy for height changes cannot be expected for slopes steeper 

 than 1:10, although model tests have shown that direction changes nearly as 

 predicted even over a vertical discontinuity (Wiegel and Arnold, 1957). 

 Accuracy where orthogonals bend sharply or exhibit extreme divergence or con- 

 vergence is questionable. This phenomenon has been studied by Beitinjani and 

 Brater (1965), Battjes (1968), and Whalin (1971). Where two orthogonals meet, 

 a caustic develops. A caustic is an envelope of orthogonal crossings caused 

 by convergence of wave energy at the caustic point. An analysis of wave 

 behavior near a caustic is not available; however, qualitative analytical 

 results show that wave amplitude decays exponentially away from a caustic in 

 the shadow zone, and that there is a phase shift of it/2 across the caustic 

 (Whalin 1971). Wave behavior near a caustic has also been studied by Pierson 

 (1950), Chao (1970), and others. Little quantitative information is available 

 for the area beyond a caustic. 



h. Refraction of Ocean Waves . Unlike monochromatic waves, actual ocean 

 waves are complicated. Their crest lengths are short; their form does not 

 remain permanent; and their speed, period, and direction of propagation vary 

 from wave to wave. 



Pierson (1951), Longuet-Higgins (1957), and Kinsman (1965) have suggested 

 a solution to the ocean-wave refraction problem. The sea-surface waves in 

 deep water become a number of component monochromatic waves, each with a dis- 

 tinct frequency and direction of propagation. The energy or height of each 

 component in the spectrum may then be found and conventional refraction anal- 

 ysis techniques applied. Near the shore, the wave energy propagated in a par- 

 ticular direction is approximated as the linear sum of the wave components of 

 all frequencies refracted in the given direction from all the deepwater direc- 

 tional components. 



The work required from this analysis, even for a small number of indivi- 

 dual components, is laborious and time consuming. Research by Borgman (1969) 

 and Fan and Borgman (1970) has used the idea of directional spectra which may 

 provide a technique for rapidly solving complex refraction problems. 



2-74 



