^
Dnj
(σ2 - σ2 ) d 2 - σ2 + d (σ2 + σ2 )
C1nj = -
2 d 2 - σn
n
j
n
n
j
2
^
C2nj = -C1nj - Dnj (σ2 - σ2j )
(18)
n
^
C3nj = Dnj (σ2 - σ2j )
n
^
C4nj = 2dσ j Dnj
Dnj
^
Dnj =
(19)
(σn - σ2j )2 + 4d 2σ2j
2
The water-surface elevation is given by integrating Eq. 7 with Eq. 15 for u to give
J
(2n - 1)πx
N
η( x, t ) = (2n - 1) ηnj cos
(20)
L
j =1
n=1
with the real part of the following equation taken:
C2nj
C3nj
C4nj
C1nj
(eλ1t - 1) +
(eλ2t - 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
parameters: water depth, basin length, bottom friction coefficient, wind speed, and
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.
RESULTS
Example Dynamics of Analytical Solution
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 CfL =0.009, upon which a spatially uniform sinusoidal wind was
imposed with w = 10 m s-1 and CD = 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
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