which corresponds to equation 2-30. Multiply equation 2-61 by dz and 

 integrate between the limits z = -d and z = 0. The result, after 

 using the boundary conditions (eqs. 2-58 and 2-60), and neglecting 

 second-order terms, is 



l?-^l^^"'^>-^|(^d)=0. (2-62) 



Introducing the relations from equation 2-56, the following field equation 

 for the water surface displacements is obtained. 



One of the boundary conditions to be associated wit?i the field equation 

 is the input along a given input line 



T? =77j (t); X = Xp y = Yj . (2-64) 



The boundary condition associated with the coastal boundary could be of 

 two tyj)es. If the waters are limited by fixed vertical boundaries, then 

 on these boundaries 



.|..| = (2-6S, 



where I and m are the direction cosines of the normal to the coastline. 

 For a changing coastline, the boundary condition is somewhat complicated. 

 Let Xg, y2 be a point on the coastline and the maximum displacements in 

 X and y directions during a runup be S^i and S]^2> given by 



S,, = / Uo dt, S,^ = / v^ dt (2-66) 



f f 



j U2 dt, S,2 = / 



where U2 and V2 are the component velocities of the particle at the 

 edge line. These velocities are independent of the beach slope in the 

 case of inundations and will be identical with the particle velocities 

 of the vertical section passing through the edge of the undisturbed 

 waters. Summarizing, 



1? = Tjj; X = Xj, y = Vj (2-67) 



(2-56a) 



and 



9u 



at 



+ 



^ax 







9v 



at 



+ 



^'^- 







at2 



- 



4.{'^ 



ax 



a^V"a^/ 



(2-56b) 

 (2-63) 



42 



