292
6 Wave Transformation in Shallow Water
approach the wave movement in the water surface TJ(r, t) and the bottom
roughness c:(r) (where the depth is written as H = H 0 (1 + c:(r)), I c: I « 1)
are presented with the help of the Fourier integral. These values are distributed into wave vectors and frequencies, whereas the frequency of spectral
components describing the bottom unevenness is accepted to be equal to zero.
Due to the roughness of the bottom, wave scattering results in mean
wave field damping with a resonance character (Pelinovskiy, 1979). The wave
amplitude change is caused only by those spectral components of bottom
roughness, whose wave vectors satisfy the synchronous conditions:
(6.56)
where k is the wave vector of the incident wave; k 1 is the wave vector of
the scattered wave; and k 2 is the wave vector of the "frozen field" of bottom
roughness. In other words, the wave vectors of scattered waves satisfy the
known Bragg conditions. Non-resonance components of bottom roughness
can result in changing the wave frequency, but the damping is affected only
by resonance components.
The formula for the mean wave field damping increment 'Y (i.e. the wave
amplitude which can be written as (a) = ae't) is obtained (Pelinovskiy,
1979) using the Burre approximation and generalized (Lavrenov, 1980) for
the case of long dispersed waves (kHo < 1). The dispersion ratio is taken as
w 2 = gH0 /(1 + 1/3k 2 H5). The formula for the damping increment is written
as follows:
2n
nylgHOk 3
j 2 ( . ()
M = Re 1 = -
cos B 2)1 + 1/3k2 H5
2
0
3n + ()) dB
2
'
(6.57)
where
the isobath and the wave propagation direction being equal to n/2- (3, the
two-dimensional spectrum of the roughness can be written as:
So, the integral (6.57) can be estimated analytically, obtaining that:
nylgHOk 2
F(O) + F(2k cos (3) cos 2 (2,8)
f-l = - 2J1 + 1/3k2 H5
cos(/3)
(6.58)
In the case of normal wave propagation (/3 = 0), the damping is determined by the components of the bottom roughness spectrum at two wave
numbers k2 = 0 and k2 = 2k. The first one corresponds to forward scattering, whereas the second to backward scattering. If the following ratio
F(O) = F(2k) is fulfilled, wave scattering is absent in spite of the bottom
6 Wave Transformation in Shallow Water
approach the wave movement in the water surface TJ(r, t) and the bottom
roughness c:(r) (where the depth is written as H = H 0 (1 + c:(r)), I c: I « 1)
are presented with the help of the Fourier integral. These values are distributed into wave vectors and frequencies, whereas the frequency of spectral
components describing the bottom unevenness is accepted to be equal to zero.
Due to the roughness of the bottom, wave scattering results in mean
wave field damping with a resonance character (Pelinovskiy, 1979). The wave
amplitude change is caused only by those spectral components of bottom
roughness, whose wave vectors satisfy the synchronous conditions:
(6.56)
where k is the wave vector of the incident wave; k 1 is the wave vector of
the scattered wave; and k 2 is the wave vector of the "frozen field" of bottom
roughness. In other words, the wave vectors of scattered waves satisfy the
known Bragg conditions. Non-resonance components of bottom roughness
can result in changing the wave frequency, but the damping is affected only
by resonance components.
The formula for the mean wave field damping increment 'Y (i.e. the wave
amplitude which can be written as (a) = ae't) is obtained (Pelinovskiy,
1979) using the Burre approximation and generalized (Lavrenov, 1980) for
the case of long dispersed waves (kHo < 1). The dispersion ratio is taken as
w 2 = gH0 /(1 + 1/3k 2 H5). The formula for the damping increment is written
as follows:
2n
nylgHOk 3
j 2 ( . ()
M = Re 1 = -
cos B 2)1 + 1/3k2 H5
2
0
3n + ()) dB
2
'
(6.57)
where
the isobath and the wave propagation direction being equal to n/2- (3, the
two-dimensional spectrum of the roughness can be written as:
So, the integral (6.57) can be estimated analytically, obtaining that:
nylgHOk 2
F(O) + F(2k cos (3) cos 2 (2,8)
f-l = - 2J1 + 1/3k2 H5
cos(/3)
(6.58)
In the case of normal wave propagation (/3 = 0), the damping is determined by the components of the bottom roughness spectrum at two wave
numbers k2 = 0 and k2 = 2k. The first one corresponds to forward scattering, whereas the second to backward scattering. If the following ratio
F(O) = F(2k) is fulfilled, wave scattering is absent in spite of the bottom