5. Refracted Waves and Caustics. 



When very long waves such as tsunamis arrive at a shoreline, a 

 substantial amount of wave energy will be reflected seaward from the 

 shoreline. These reflected waves will interact with the bottom topogra- 

 phy, and will refract as they travel seaward. Refraction diagrams of 

 these waves show a tendency for the waves to turn parallel to the shore- 

 line as they move into deeper water. When a shelf slopes away from the 

 shoreline, and extends a sufficient distance seaward, the waves may be 

 turned back shoreward (see Fig. 35) . The line tangent to the wave rays 

 where they turn shoreward is a caustic. The wave rays will not cross 

 the caustic, and the wave energy tends to be trapped, although some wave 

 energy will leak across the caustic (Chao, 1970; Chao and Pierson, 1970; 

 Pierson, 1972). 



Shoreline .^^^ 



Figure 35. Schematic of caustic (uniform bottom slope). 



Chao (1970) and Chao and Pierson (1970) investigated higher frequency 

 waves trapped by a caustic. They demonstrate that lower frequency (longer 

 period) waves will form caustics closer to the coastline, and that waves 

 with frequencies above some maximum value will propagate seaward into 

 deep water. For tsunamis, only the lower frequency waves are significant. 



As the wave rays are not normal to the shoreline, different parts of 

 the wave crest would arrive at the shoreline at different times. Where a 

 coastline is irregular, parts of a wave crest reflected from one section 

 of coastline may be refracted and trapped so that they coincide with an 

 incident wave on another section of coastline. Palmer, Mulvihill, and 

 Funasaki (1965) illustrated the effects of wave trapping at Hilo, Hawaii, 

 where the reflected wave rays were turned by refraction so that they 

 arrived simultaneously at a point inside Hilo Harbor (see Fig. 36). 



The case of wave energy being trapped by refraction can be most 

 easily illustrated for a long, shallow-water wave on a straight section 

 of shoreline, with some water depth, d g , at the toe of the shoreline 

 slope, and with a constant shelf slope extending seaward. It is assumed 

 that the wave reflects from the shoreline slope and refracts on the shelf. 



108 



