^

(σ2 - σ2 ) *d * 2 - σ2 + *d *(σ2 + σ2 )

2 *d * 2 - σn

2

^

(18)

^

^

^

(19)

(σn - σ2*j *)2 + 4*d * 2σ2*j*

2

The water-surface elevation is given by integrating Eq. 7 with Eq. 15 for *u *to give

J

(2*n *- 1)π*x *

η( *x*, *t *) = (2*n *- 1) ηnj cos

(20)

j =1

with the real part of the following equation taken:

(*e*λ1t - 1) +

(*e*λ2t - 1) +

sin σ jt -

(cos σ jt - 1)

ηnj =

(21)

λ1

λ2

σj

σj

This solution describes linearized wind-forced motion in a 1D basin as governed by five

fundamental frequency of the oscillatory wind. The solution includes the initial transients

and possible mixed under-damping (*d *< σn) and over-damping (*d *> σn), depending on the

normal modes, as can occur according to the values of λ1 and λ2.

For examining properties of the analytic solution, the geometry of an idealized basin

was established that approximated Baffin Bay, Texas (Militello and Kraus 2001), as *L *=

29 km, *h *= 1 m, and *C*fL =0.009, upon which a spatially uniform sinusoidal wind was

imposed with *w *= 10 m s-1 and *C*D = 0.0016. Through trial runs, 3-digit reproducibility was

obtained with four wind harmonics (*J *= 4) and nine normal modes (*N *= 9).

basin are plotted in Fig. 2 for 3 days. Day 0 was omitted to allow transients to disappear.

The greatest variation in water level and velocity are experienced at the basin ends and

middle, respectively.

The water-surface elevation and current velocity contain complex structure through the

presence of both wind-generated harmonics and normal-mode frequencies. The spectra of

η and *u *shown in Fig. 3 indicate strong motion at the fundamental frequency of 1 cpd and

energy at the odd forced harmonics associated with the quadratic wind stress. The peak at

4.65 cpd is the first normal (seiching) mode of the basin. The amplitudes of the harmonics

can also be obtained from the solutions, and Eqs. 15 and 20.

6

Kraus & Militello