Murty (1977) indicates that the value of (L/d) 2 is a measure of 

 frequency dispersion; the value of H/d is a measure of amplitude 

 dispersion. Murty points out that the linear long-wave equations are 

 valid when U << 1. In this case H/d is small and amplitude dispersion 

 can be ignored. 



When U is of the order one [U = 0(1)], both amplitude and frequency 

 dispersion are important. In this case Boussinesq or Korteweg-deVries 

 equations should be used. Where U >> 1 amplitude dispersion dominates 

 the solution, and the finite-amplitude, nonlinear long-wave equations 

 should be used. It should be emphasized that when U = 0(1) it is not 

 necessary that 11=1. Zabusky and Galvin (1971) show that the Korteweg- 

 deVries equation accurately describes wave propagation for U < 800 in 

 some cases. 



For tsunamis with very long periods (and therefore long wavelengths) , 

 the condition that U << 1 is usually never satisfied. However, the error 

 which results from the use of the linear equations is quite small as long 

 as the value of H/d is small. The acceptable limit of the value for 

 H/d (i.e., the point where the error in the calculations becomes signif- 

 icant) depends in part on the rate of shoaling of the wave, i.e., the 

 shoreward slope of the bottom topography. 



****** 



****** EXAMPLE PROBLEM 3********** 



GIVEN : A tsunami has a period of 20 minutes and a wave height of 0.05 

 meter (0.16 foot) in a 1,000-meter (3,280 feet) water depth. 



FIND : The parameter U. 



1 



SOLUTION : From equation (31) , 



C = /gd = /9.807 x 1,000 = 99 meters (325 feet) per second 

 L = CT = 99 x 20 x 60 = 118,800 meters 



From equation (65) , 



rj = M / L\ 2 _ 0.05 / 118,800 \ 2 

 d \d/ 1,000 \ 1,000 / 



U = 0.706 

 ************************************* 



In more recent investigations, the parameter U given by equation 

 (65) has been redefined as U* where 



46 



