and satisfies the Laplacian 



^.ff.^=0 C2-30, 



dx'^ dy^ 9z^ 



at every point in the region of the model or the prototype. At the solid 

 boundaries 



^.-0 C2-31a) 



i.e., the velocity component normal to the solid boundary vanishes. Let 

 ^o> Yo) Zq be a point on the solid boundary and £, m, n be the direction 

 cosines of the normal to the boundary at this point. Equation 2-31a may 

 now be written as 



£^ + m^ + n^-0. (2-31b) 



ox ay az ^ -^ 



If the equation of the solid boundary is in the form 



f(x,y,z) = , (2-32) 



an alternate expression to equation 2-31b is 



K^ + K^ + KM = Q (-2-331 



9x 9x 9y 9y 9z 9z ' *• ■' 



The velocity potential needs to satisfy two boundary conditions, the 

 dynamic and the kinematic. Assuming that the pressure over the water 

 surface is constant, and equal to zero, then the dynamic surface condi- 

 tion takes the form 



1^ + g7? = 0, z = . (2-34) 



dt 



The kinematic surface condition, assuming that products such as u(3h/9x) 

 can be ignored, takes the form 



|f*|a=0, z = (2-35) 



where n is the surface elevation, measured from the imdisturbed water 

 level, and the z-axis is drawn vertically upward. The conditions for 

 similarity between model and prototype can be readily obtained after the 



37 



