Geoscience Reference
In-Depth Information
velocity, therefore, the problem of wave damping under the action of bottom friction
is certainly not linear, and, consequently, one can expect the damping not to be
exponential in character.
For definiteness, we shall consider the one-dimensional problem of a sine wave
propagating in the positive direction of axis 0
x
in a basin of fixed depth
H
,
−
g
Ht
)
.
u
(
x
,
t
)=
u
0
sin(
x
Assume the action of the friction force to be insignificant, so that the amplitude
and shape of a wave covering a distance comparable with the wavelength
undergo
no significant changes. We shall estimate the wave energy per period in space (and
per unit length of front), as twice the kinetic energy,
λ
λ
u
0
u
2
d
x
=
ρ
H
λ
E
=
ρ
H
.
(5.20)
2
0
Strictly speaking, such an estimate is only valid for linear waves, but, as we have
already noted, we consider the non-linearity to be weak.
The losses of wave energy per unit time in the region from 0 up to
, that are due
to the action of bottom friction, are determined by the following formula (the point
above the variable signifies differentiation with respect to time):
λ
λ
λ
u
0
4
C
B
ρλ
E
=
3
d
x
=
(
T
B
,
u
) d
x
=
−
C
B
ρ
|
u
|
−
.
(5.21)
π
3
0
0
Now, pass in formulae (5.20) and (5.21) to specific energy per unit mass of liquid
=
u
0
E
b
≡
2
,
(5.22)
ρ
H
λ
E
4
C
B
u
0
3
b
≡
=
−
.
(5.23)
ρ
H
λ
π
H
Excluding the quantity
u
0
in expression (5.23), with the aid of the constraint
(5.22) one obtains the ordinary differential equation
8
√
2
C
B
b
3
/
2
3
b
=
−
.
(5.24)
π
H
We recall, that we are tracing the energy of a sole space period of the wave. The
ordinary differential equation (5.24) describes the variation of this quantity in time.
The solution of equation (5.24), written with respect to the wave velocity ampli-
tude
u
0
, has the following form:
