follows, the full interval [0, L] is chosen as the spatial domain, with symmetry about L/2 for
u and anti-symmetry for η serving as checks of the solution.
A solution is sought of the form of a Fourier expansion
(2n - 1)π x
N
u( x, t ) = ∑ un (t ) sin
(11)
L
n=1
which is a normal-mode equation satisfying the lateral boundary conditions. Substitution of
Eq. 11 into Eq. 10 shows that the un satisfy the equation describing forced motion with
damping,
N
(un )tt + 2d (un )t + σ u = ∑ Fn
2
(12)
nn
n=1
with σn = (2n-1)πc/L corresponding to odd normal modes. Eqs. 3, 9, and 10 give
N
Fn = ∑ Dnj cos(σ jt )
(13)
j =1
4
ρ σ
Dnj =
CD a w2 Aj
(14)
(2n - 1)π
ρ h
The solution of Eq. 12 with the initial conditions depends on the relative values of σn
d contains the water depth and that the σn will have a wide range if a reasonable number of
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(σt) 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
(2n - 1)πx
N
u( x, t ) = ∑ ∑ unj sin
(15)
L
n=1 j =1
with the real part of
unj = C1nj eλ1t + C2nj eλ2t + C3nj cos σ jt + C4nj sin σ jt
(16)
λ1 = -d + d 2 - σn
2
(17)
λ2 = -d - d - σ
2
2
,
n
5
Kraus & Militello
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