K* 



1.4 



t 

 1.2 



1.0 



0.8 



0.6 



K r 



0.4 



0.2 



0.45 0.50 



0.60 



d 2 



0.70 

 1/2 



0.80 



0.90 



Figure 23. Reflection and transmission coefficients 

 (modified from Bourodimos and Ippen, 196? 



5. Solitons and Shoaling- Induced Dispersion . 



For certain conditions, a wave will decompose into a train of waves. 

 Examples of this are shown in Figures 24 to 28. This train of waves will 

 consist of an initial wave having the highest amplitude, followed by a 

 finite number of waves of decreasing amplitude. Wave decomposition has 

 been investigated by Mason and Keulegan (1944), Horikawa and Wiegel (1959), 

 Benjamin and Feir (1967), Street, Burgess, and Whitford (1968), Madsen and 

 Mei (1969), Byrne (1969), Street, Chan, and Fromm (1970), Galvin (1970), 

 Zabusky and Galvin (1971), and Hammack and Segur (1974). 



Benjamin and Feir (1967) discuss the stability of waves, and indicate 

 that the waves will only be unstable if kd > 1.363, where k is the wave 

 number 2ir/L. Whitham (1967) showed that equations governing extremely 

 gradual variations in wave properties are elliptic if kd > 1.363, and 

 hyperbolic if kd < 1.363. For tsunamis, where d/L << 1, the equations 

 will be hyperbolic and the waves will be stable, at least in a constant 

 water depth. 



80 



