120
3. Beyondthe One-Way Wave Equation
real values of ware obtained whenever
max {IJ.LI , lvI} s 1.
(3.27)
As before, suppose that U = e eos 0, V = e sin 0,
=
= Ss, and that C
is abound on [c]; then requiring striet inequality in (3.27) to guarantee that the
leapfrog time differenee does not admit weakly unstable modes, the stability condition becomes C
case.
I < 1, which is identical to that for the one-dimensional
Although the averaging scheme is potentially more efficient because it permits
longer time steps, it is also less accurate. This loss of aecuracy is not clearly
refleeted in the truneation error, which is 0
+ 0
for both the
nonaveraged method (3.21) and averaging seheme (3.24) . The problems with the
averaging scheme appear in the representation of the poorly resolved waves. As
discussed in eonneetion with Fig. 3.3, shorter waves are resolvable on a twodimensional grid, and if properly represented by the spatial differencing, they
should generate higher-frequency oseillations and reduce the maximum stable
time step. The averaging seheme avoids such time-step reduction by artificially
reducing the phase speeds of the diagonally propagating waves.
The phase-speed errors introduced by the spatial differencing in both methods
may be examined by a generalization of the one -dimensional approach discussed
in Section 2.4.1 . First eonsider the propagation of two-dimensional waves in the
nondiscretized problem. Waves of the form
""(x, y, t) = e i(kxHy-wt)
satisfy the two-dimensional adveetion equation (3.20), prov ided that
w=v ·k,
(3.28)
where v is the velocity vector and k is the wave number vector with eomponents
(k , l). If K denotes the magnitude ofk, then the x and y eomponents ofthe wave
number veetor may be expressed as
k = K cos 0 and l = K sin 0,
(3.29)
where 0 is the angle between the wave number vector and the x-axis.The dispersion relation (3.28) implies that all apparent wave propagation is parallel to the
wave number veetor. Consider, therefore, the case in which the velocity veetor is
parallel to the wave number veetor. Then if e is the wind speed,
U = eeosO and V = esinO.
(3.30)
Substituting (3.29) and (3.30) into the dispersion relation demonstrates that e =
os]K, implying that the phase speed is equal to the wind speed.
As in the one-dimensional case, the phase speeds of the waves generated by
the finite-differenee approximations (3.21) and (3.24) are defined as c" = io" I K,
3. Beyondthe One-Way Wave Equation
real values of ware obtained whenever
max {IJ.LI , lvI} s 1.
(3.27)
As before, suppose that U = e eos 0, V = e sin 0,
=
= Ss, and that C
is abound on [c]; then requiring striet inequality in (3.27) to guarantee that the
leapfrog time differenee does not admit weakly unstable modes, the stability condition becomes C
case.
I < 1, which is identical to that for the one-dimensional
Although the averaging scheme is potentially more efficient because it permits
longer time steps, it is also less accurate. This loss of aecuracy is not clearly
refleeted in the truneation error, which is 0
+ 0
for both the
nonaveraged method (3.21) and averaging seheme (3.24) . The problems with the
averaging scheme appear in the representation of the poorly resolved waves. As
discussed in eonneetion with Fig. 3.3, shorter waves are resolvable on a twodimensional grid, and if properly represented by the spatial differencing, they
should generate higher-frequency oseillations and reduce the maximum stable
time step. The averaging seheme avoids such time-step reduction by artificially
reducing the phase speeds of the diagonally propagating waves.
The phase-speed errors introduced by the spatial differencing in both methods
may be examined by a generalization of the one -dimensional approach discussed
in Section 2.4.1 . First eonsider the propagation of two-dimensional waves in the
nondiscretized problem. Waves of the form
""(x, y, t) = e i(kxHy-wt)
satisfy the two-dimensional adveetion equation (3.20), prov ided that
w=v ·k,
(3.28)
where v is the velocity vector and k is the wave number vector with eomponents
(k , l). If K denotes the magnitude ofk, then the x and y eomponents ofthe wave
number veetor may be expressed as
k = K cos 0 and l = K sin 0,
(3.29)
where 0 is the angle between the wave number vector and the x-axis.The dispersion relation (3.28) implies that all apparent wave propagation is parallel to the
wave number veetor. Consider, therefore, the case in which the velocity veetor is
parallel to the wave number veetor. Then if e is the wind speed,
U = eeosO and V = esinO.
(3.30)
Substituting (3.29) and (3.30) into the dispersion relation demonstrates that e =
os]K, implying that the phase speed is equal to the wind speed.
As in the one-dimensional case, the phase speeds of the waves generated by
the finite-differenee approximations (3.21) and (3.24) are defined as c" = io" I K,