d. The fluid is inviscid. Thus, internal forces due to viscosity 

 are neglected. 



e. The seabed is regarded as fixed and impermeable. 



f. The effects of surface waves are considered linearly super- 

 imposable on the storm surge. 



Thus, it is assumed that only horizontal flow takes place, the 

 traditional approach when dealing with this type of fluid motion. Such 

 flows have been referred to as neaply horizontal flows (Birkhoff , 1960) . 

 In the context of wave motions, such wave motions are often referred to 

 as waves of long -period or simply long waves. Wave motions in which the 

 vertical accelerations have a marked influence on the wave behavior are 

 called short-period waves or gravity waves. Surface waves referred to 

 in item f above are of this type. 



The water motion which accompanies the propagation of long waves is 

 unsteady, and is in a continuous state of change. This change, however, 

 is not abrupt, and the motion can be considered as gradually changing. 



2. Basic Notation 



Figure 1 shows the various notations used in the discussions. The 

 value D is the total depth of the fluid at time t. Moreover, D = S + d 

 where S is the disturbance height of the free surface, and d is the depth 

 of the undisturbed fluid. The Cartesian axes x and y are situated in a 

 horizontal reference plane at the undisturbed water level, with z directed 

 vertically upwards. The x and y axes are chosen counterclockwise with x 

 directed shoreward and perpendicular to the shoreline. 



3. The Differential Equations 



The hydrodynamic equations may be written in either of the two 

 equivalent forms: (1) the mean current velocities and (2) the volume 

 transport. Preference for the particular form depends on the individual 

 investigator, but generally the second form is preferred when an elec- 

 tronic computer carries out the computations. Here the volume- transport 

 form is taken. This form is obtained by integrating the governing equa- 

 tions in the vertical over the total depth. Integrations of the primitive 

 equations appropriate for the storm surge problem have been shown by 

 Haurwitz (1951), Weylander (1961), Fortak (1962), Platzman (1963), Reid 

 (1964), and Harris (1967). These derivations show more clearly the actual 

 approximations involved. Here, the equations are taken directly in 

 integrated form, since it is the purpose co display only the principal 

 approximations taken. 



The governing two-dimensional equations are: 



MM T — X 



iU ^ JS£ , .JSI = fv - gD ^ + gD ^ + ^ ^ - W P (1) 



3t 8x 9y ^ 3x ^ 9x p x 



