7.6. IRREGULAR (2-D) WAVE GENERATION
409
with
1 [+to
an = —
Tj(xo,r)cos(knxo -unr)dr
(7.200)
lO J-to
1 f
bn = —
7/(ro, r) sin(Arnx0 - u>nr)dr
(7.201)
J-to
and
kn
Un
t
T
- wavenumber of the nth component
- radian frequency of the nth component
time
- dummy variable of integration
Under the assumption that all components propagate independently
from the wave board (located for convenience at x = 0) the sea surface at
the wave board can be written as
which can be expressed in terms of Fourier amplitudes and phases as
oo
q(0,/) = 52 cos(wn<) - 6nsin(wn0]
(7.202)
n=l
By noting that the wave board displacement for a single wave component
is given as
OO
7/(0, t) = 52 Cn COs(wn* + 0n)
(7.203)
n = l
and from Eqn. 7.193 the component wave amplitude is given in terms of
wave board amplitude as cn = (mi„) e> we can transform the amplitudes
and phases in Eqn. 7.203 to give an equivalent time series for wave board
displacement written as
X(t) = e sin(a?t + 0)
(7.204)
where the tf/2 phase shift has the effect of changing the cosine into sine
After manipulation the time series becomes
OO
X(t) = 52
c-os(unt + 0n - tt/2)
(7.205)
n=l min
OO
-
X(i) = Y' ----- [a„ sin(WnC + bn cos(w„t)]
(7.206)
mi-