η

Fig. 1. Sketch of one-dimensional basin with wind forcing, *t *=0

∂η

∂*u*

= -*h*

(7)

∂*t*

∂*x*

and

∂η *C * fL

∂*u*

= -*g * -

(8)

∂*t*

∂*x*

where the wind forcing is represented by the function

ρa W W

(9)

ρ* h*

for pure oscillatory wind specified by Eq. 1 with *w*0 = 0. Although the wind-drag

coefficient varies with the wind speed in some formulations, it is taken to be constant for

this derivation, as is *C*fL.

From Lamb (1945), Ippen and Harleman (1966), and others, linear equation systems

such as Eqs. 7 and 8 can be solved by differentiating Eq. 7 with respect to *x *and Eq. 8 with

respect to *t*, then adding the resultant equations to eliminate η. The one-dimensional

inhomogeneous wave equation for *u *is obtained,

(10)

in which notation was simplified by defining *d *= *C*fL/(2*h*), and where *c*2 = *gh*. The

subscripts denote partial differentiation with respect to *t *and *x*. The quantity *d *has the

with the depth.

For the idealized basin, the initial and boundary conditions on *u *are, respectively,

horizontal, and the wind begins blowing at *t *= 0. Symmetry indicates that the problem can

be solved over half the basin, for example, on [0, *L*/2]. In the solution procedure that

4

Kraus & Militello