follows, the full interval [0, *L*] is chosen as the spatial domain, with symmetry about *L*/2 for

A solution is sought of the form of a Fourier expansion

(2*n *- 1)π x

(11)

which is a normal-mode equation satisfying the lateral boundary conditions. Substitution of

Eq. 11 into Eq. 10 shows that the *u*n satisfy the equation describing forced motion with

damping,

(*u*n )tt + 2*d *(*u*n )t + σ *u *= ∑ Fn

2

(12)

with σn = (2*n*-1)π*c*/*L *corresponding to odd normal modes. Eqs. 3, 9, and 10 give

(13)

4

ρ σ

(14)

(2*n *- 1)π

ρ* h*

The solution of Eq. 12 with the initial conditions depends on the relative values of σn

components (e.g., *N *= 7) is assigned. Critical damping (*d *= σn) cannot occur in a practical

situation for input values specified to one or two significant figures. The formal solution

given below for overdamping describes both the under- and overdamping situations for

complex arguments of the exponential functions appearing in it.

The solution of the linearized shallow-water wave equations for the basin with an

impressed wind blowing as *W *= *w *sin(σ*t*) and with initial conditions of a flat water surface

and boundary conditions of zero velocity is found to be, for the depth-averaged velocity,

J

(2*n *- 1)π*x *

(15)

with the real part of

(16)

λ1 = -*d *+ *d * 2 - σn

2

(17)

λ2 = -*d *- *d *- σ

2

2

,

5

Kraus & Militello

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