wave height, for an arrival time equal to the calculated time for the 

 reflected wave, was in good agreement with the calculated wave height. 



Figure 17 shows that for waves arriving with a higher angle of 

 incidence there will be some value of d 1 /d 2 for which equation (163) 

 predicts no reflected wave. For these same values of d 1 /d 2 , the pre- 

 dicted transmitted wave height would equal the incident wave height. 

 Cochrane and Arthur note that reflected waves are normally of secondary, 

 but not negligible, magnitude according to theory. At given stations, 

 convergence may cause reflected waves to be of primary magnitude, but 

 this occurs only in relatively few cases. Shepard, MacDonald, and Cox 

 (1950) note that the highest and most damaging waves at Napoopoo and 

 Hokeena, on the island of Hawaii, from the 1 April 1946 tsunami origi- 

 nating in the Aleutian Islands, Alaska, were reflected waves from the 

 continental slopes of Japan and the Bonin Islands. 



2. Linear Depth Transitions . 



Cochrane and Arthur (1948) indicate that the length of a continental 

 slope, as well as the difference in water depth, should be considered in 

 the calculations. Dean (1964) considered the case of a wave normally 

 incident on a linear change in water depth shown in Figure 19. Defining 

 a parameter, Zj , as 



Z l 



4-rrd 



(172) 



the transmission coefficient 



and the reflection coefficient 



h = hT (173 ) 



i 



H 



K --.:-£ 

 * H. 



(174) 



Dean found the results shown in Figure 20. As before, for the zero angle 

 of incidence assumed by Dean, when dj/dj < 1.0 the value of the reflec- 

 tion coefficient K is negative. When d 1 /d 2 > 1.0 the value of K^ is 



positive. 



EXAMPLE PROBLEM 5 



GIVEN : An incident wave, which is 0.5 meter (1.64 feet) high and has a 

 period of 40 minutes, recedes from the coastline through water 100 

 meters deep and passes from the shallow water over a shelf into water 

 3,025 meters (9,925 feet) deep. The transition between the two water 



73 



