70
4 Long Waves in a Channel
It can also be shown (Cushman-Roisin, 1994) that these waves are governed by
the well-known dispersion relation:
c =
λ
T
=
gh
(4.16)
implying that the phase speed of a long surface gravity wave exclusively depends on
total water depth. Consequently, it follows that the ratio between horizontal speed
of a water parcel and phase speed is very small:
u o
c
=
η
h
<< 1
which is justificatio for neglection of the nonlinear terms. For instance, a long wave
of 1 m in amplitude in a 100 m deep ocean propagates with a phase speed of about
c = 30 m/s, while water parcels attain maximum lateral displacement speeds of only
u o = 0.3 m/s.
Horizontal fl w under a long surface wave is depth-independent and so are horizontal gradients of u. On the basis of the local form of the continuity equation, given
for our channel by:
∂w
∂z
= −
∂u
∂ x
we can derive the solution for vertical speed of a flui parcel as a function of depth:
w(t, x, z
∗
) = −2π u o z
∗
/λ cos (2π x/λ − 2π t/T )
where z
∗
is (positive) distance from the seafloo . Vertical speed vanishes at the plane
seafloo (per definition and approaches an oscillating maximum at the sea surface.
The ratio between vertical and horizontal speeds of water parcels is 2π h/λ. This
ratio is small compared with unity for shallow-water waves (λ >> h). Accordingly,
motions of water parcels in a shallow-water wave are largely horizontal. Another
important feature inherent with long waves is that they reach the seafloo and are
capable of stirring up sediment from the seafloo , if energetic enough. Figure 4.4
shows a snapshot of the analytical solution of a shallow-water wave.
4.2.6 Animation Script
A SciLab script, called “AnalWaveSol.sce”, can be found in the folder “Miscellaneous/Waves” on the CD-ROM accompanying this book. This script creates an
animation of the analytical wave solution.
4 Long Waves in a Channel
It can also be shown (Cushman-Roisin, 1994) that these waves are governed by
the well-known dispersion relation:
c =
λ
T
=
gh
(4.16)
implying that the phase speed of a long surface gravity wave exclusively depends on
total water depth. Consequently, it follows that the ratio between horizontal speed
of a water parcel and phase speed is very small:
u o
c
=
η
h
<< 1
which is justificatio for neglection of the nonlinear terms. For instance, a long wave
of 1 m in amplitude in a 100 m deep ocean propagates with a phase speed of about
c = 30 m/s, while water parcels attain maximum lateral displacement speeds of only
u o = 0.3 m/s.
Horizontal fl w under a long surface wave is depth-independent and so are horizontal gradients of u. On the basis of the local form of the continuity equation, given
for our channel by:
∂w
∂z
= −
∂u
∂ x
we can derive the solution for vertical speed of a flui parcel as a function of depth:
w(t, x, z
∗
) = −2π u o z
∗
/λ cos (2π x/λ − 2π t/T )
where z
∗
is (positive) distance from the seafloo . Vertical speed vanishes at the plane
seafloo (per definition and approaches an oscillating maximum at the sea surface.
The ratio between vertical and horizontal speeds of water parcels is 2π h/λ. This
ratio is small compared with unity for shallow-water waves (λ >> h). Accordingly,
motions of water parcels in a shallow-water wave are largely horizontal. Another
important feature inherent with long waves is that they reach the seafloo and are
capable of stirring up sediment from the seafloo , if energetic enough. Figure 4.4
shows a snapshot of the analytical solution of a shallow-water wave.
4.2.6 Animation Script
A SciLab script, called “AnalWaveSol.sce”, can be found in the folder “Miscellaneous/Waves” on the CD-ROM accompanying this book. This script creates an
animation of the analytical wave solution.