where 



the radius of the Earth 



degrees latitude (measured from the pole) 



degrees longitude (measured eastward) 



the velocity in the 6-direction 



the velocity in the ^-direction 



time 



the coriolis parameter = 2 fi cos 6 



the rotational speed of the Earth in radians per second 



Figure 9. Spherical coordinate system. 



For linear long-wave equations, the acceleration component in the radial 

 direction is considered to be negligible. The continuity equation is 



-^ = : (— [(d + n) u sin 9] + i- [(d + n) v]| + 21 



3t R sin 6 be 3* i 3t 



(78) 



where r, is a time-dependent vertical bottom displacement in the generat- 

 ing area and equal to zero elsewhere. 



For instantaneous bottom displacement, the initial wave is assumed to 

 be a water surface displacement equivalent to the bottom displacement, 

 and 3?/3t =0. If a wave is generated near a coastline, i.e., in rela- 

 tively shallow water, the parameter U may initially be quite large. 

 For a wave propagating away from the coastline, it is assumed that the 



50 



