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I. Gankevich and A. Degtyarev
that ACF represented by exponentially decaying cosines satisfies first order Stokes’
equations for gravity waves [17]. So, if the shape of the wave profile is the only
concern in the simulation, then one can simply multiply it by a decaying exponent to
get appropriate ACF. This ACF does not reflect other wave profile parameters, such
as wave height and period, but opens possibility to simulate waves of a particular
non-analytic shape by “drawing” their profile, then multiplying it by an exponent
and using the resulting function as ACF. So, this empirical method is imprecise but
offers simpler alternative to Wiener–Khinchin theorem approach; it is mainly useful
to test ARMA model.
Standing Wave ACF
For three-dimensional plain standing wave the profile is given by
í µí¼ (t, x, y) = A sin(k x x + k y y) sin(í µí¼t).
(9)
Find ACF via analytic method. Multiplying the formula by a decaying exponent
(because Fourier transform is defined for a function f that f ⟶
x→±∞
0) yields
í µí¼ (t, x, y) = A exp
[ −í µí»¼(|t| + |x| + |y|)
] sin(k x x + k y y) sin(í µí¼t).
(10)
Then, apply 3D Fourier transform to the both sides of the equation via symbolic
computation programme, fit the resulting polynomial to the following approximation:
K(t, x, y) = í µí»¾ exp
[ −í µí»¼(|t| + |x| + |y|)
] cos í µí»½t cos
[ í µí»½x + í µí»½y
] .
(11)
So, after applying Wiener–Khinchin theorem we get initial formula but with cosines
instead of sines. This difference is important because the value of ACF at (0, 0, 0)
equals to the ARMA process variance, and if one used sines the value would be
wrong.
If one tries to replicate the same formula via empirical method, the usual way is
to adapt (10) to match (11). This can be done either by changing the phase of the
sine, or by substituting sine with cosine to move the maximum of the function to the
origin of coordinates.
Propagating Wave ACF
Three-dimensional profile of plain propagating wave is given by
í µí¼ (t, x, y) = A cos(í µí¼t + k x x + k y y).
(12)
For the analytic method repeating steps from the previous two paragraphs yields
K(t, x, y) = í µí»¾ exp
[ −í µí»¼(|t| + |x| + |y|)
] cos
[ í µí»½(t + x + y)
] .
(13)
I. Gankevich and A. Degtyarev
that ACF represented by exponentially decaying cosines satisfies first order Stokes’
equations for gravity waves [17]. So, if the shape of the wave profile is the only
concern in the simulation, then one can simply multiply it by a decaying exponent to
get appropriate ACF. This ACF does not reflect other wave profile parameters, such
as wave height and period, but opens possibility to simulate waves of a particular
non-analytic shape by “drawing” their profile, then multiplying it by an exponent
and using the resulting function as ACF. So, this empirical method is imprecise but
offers simpler alternative to Wiener–Khinchin theorem approach; it is mainly useful
to test ARMA model.
Standing Wave ACF
For three-dimensional plain standing wave the profile is given by
í µí¼ (t, x, y) = A sin(k x x + k y y) sin(í µí¼t).
(9)
Find ACF via analytic method. Multiplying the formula by a decaying exponent
(because Fourier transform is defined for a function f that f ⟶
x→±∞
0) yields
í µí¼ (t, x, y) = A exp
[ −í µí»¼(|t| + |x| + |y|)
] sin(k x x + k y y) sin(í µí¼t).
(10)
Then, apply 3D Fourier transform to the both sides of the equation via symbolic
computation programme, fit the resulting polynomial to the following approximation:
K(t, x, y) = í µí»¾ exp
[ −í µí»¼(|t| + |x| + |y|)
] cos í µí»½t cos
[ í µí»½x + í µí»½y
] .
(11)
So, after applying Wiener–Khinchin theorem we get initial formula but with cosines
instead of sines. This difference is important because the value of ACF at (0, 0, 0)
equals to the ARMA process variance, and if one used sines the value would be
wrong.
If one tries to replicate the same formula via empirical method, the usual way is
to adapt (10) to match (11). This can be done either by changing the phase of the
sine, or by substituting sine with cosine to move the maximum of the function to the
origin of coordinates.
Propagating Wave ACF
Three-dimensional profile of plain propagating wave is given by
í µí¼ (t, x, y) = A cos(í µí¼t + k x x + k y y).
(12)
For the analytic method repeating steps from the previous two paragraphs yields
K(t, x, y) = í µí»¾ exp
[ −í µí»¼(|t| + |x| + |y|)
] cos
[ í µí»½(t + x + y)
] .
(13)