Environmental Engineering Reference
In-Depth Information
Figure 21: Range of application of various wave theories.
certain depth and wave steepness conditions. It can be seen that linear Airy wave
theory can be applied in deep water waves with small steepness. Beyond this
region non-linear wave theories such as Stokes' 5th order and stream function
waves apply. This region in turn is limited by the wave breaking limit. In shallow
water waves cannot grow higher than 0.78 times the water depth, while in deep
water a wave will break if it grows too steep, with the wave height exceeding
0.14 times the wave length.
Linear Airy wave theory considers the surface elevation to be described by a
harmonic wave:
( 1)
h
(,)
xt
=
a
sin(
w
t
kx
)
Using potential theory and boundary conditions at the seabed and at the free
surface a velocity potential F can be formulated corresponding to the surface ele-
vation described as in the following equation:
w
a
cosh
k h
(
+
z
)
Φ=
(,,)
xzt
cos(
w
t
kx
)
(2 )
k
sinh
kh
is the exponential decay func-
tion that describes the decrease of the intensity of the kinematics with increas-
ing depth. By differentiating the velocity potential with respect to x and z the
horizontal velocity u and the vertical velocity w can be derived, respectively,
as follows:
In this equation the term cosh
kh z
(
+
) sinh
kh
cosh
kh z
(
+
)
sinh
kh z
(
+
)
( 3 )
ua
=
w
sin(
w
t
kx
),
wa
=
w
cos(
w
t
kx
)
sinh
kh
sinh
kh
The accelerations can be determined by differentiation of the horizontal and
vertical velocities with respect to t .
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