linear long-wave equations can be applied to the initial propagation, 

 and that the resulting errors are of a size that can be accepted in the 

 calculations . 



4. Nearshore Propagation . 



The linear long-wave equations may be used for the propagation of 

 waves from a shoreline, across an ocean basin, and up to an area near 

 another shoreline. It is also necessary to consider the propagation of 

 a tsunami toward a shoreline from a nearby generating area, or into the 

 nearshore area at a distant shoreline where the linearized long-wave 

 equations will not provide solutions with sufficient accuracy. Peregrine 

 (1967) derived equations for three-dimensional long waves in water of 

 varying depth (i.e., shoaling waves) which correspond to the Boussinesq 

 equations for solitary waves in water of constant depth. An expansion 

 is used similar to that used by Keller (1948) . 



The dimensional variables are defined with an *, and the dimension- 

 less variables by the following equations: 



x* y* z* 



x = — , y = — , z = — 



d d d 



o o o 



(t 



\1/ 



o 



/ Pg o 



(gd/ /2 (gd o ) 1/2 (gd o ) 1/2 



where d is a length representative of water depth; p the pressure; 

 the velocity u in the x-direction, v in the y-direction, w in the 

 z-direction; and the other variables are defined as before. Defining 



q = (u 2 + v 2 ) 1/2 and Q = f qdz (79) 



where q is velocity and Q the flow rate, the continuity equation is 



3Q x »% 3n n 



IT + iy~ + IT = ° (80) 



where Q^ is the component of Q in the x-direction, and Qj, the 

 component of Q in the y-direction. 



Euler's equations of motion are 



3q 3q 3q 3q 8p 8p 



— +u— i-+v — +w — + — +— = (81) 



3t 3x 8y 3z 3x 3y 



and 



3w 3w 3w 3w 3p -. _ , 00 . 



— +u — +v — +w — +^E-+ 1 = (82) 

 3t 3x 3y 3z 3z 



51 



