194
5 Wave Evolution in Non-uniform Currents in Deep Water
calculations. For this purpose a modified traditional scheme of the WKB approximation can be applied. In order to do so the Maslov method (Kravtsov,
1968; Maslov & Fedoryuk, 1976), describing the uniform wave field asymptotics in the entire space can be used. The Maslov method is often used in
obtaining a solution of the Helmholtz equation and it can be briefly described
as follows.
As mentioned above, the application of the geometrical optics method
leads to an amplitude value singularity in the caustic. However, when the
wave field is described by the integral representation the wave amplitude in
the caustic is finite in value. The Fourier integral can be used as the representation. Integration is with respect to the wave vector component k,
in which the direction the wave medium properties are changed (for example, along the Ox axis). In other words, one proceeds from the spatial coordinates { x1 = x, x2 = y} to the mixed spatial-momentum representation
{ k1 = kx, x 2 = y}. If the medium properties are changed in two directions
simultaneously, it is necessary to proceed from the spatial to the momentum
coordinates, i.e., to find a solution as the Fourier double integral.
However, the formal application of this method for waves in non-uniform
currents would be wrong, if the waves were described by a more complicated
equation than the Helmholtz equation (5.45), for which the aforementioned
asymptotic methods were initially developed. In this case the wave action
density is preserved rather than the energy. Voronovich and Goncharov (1982)
proposed the idea of generalizing the Maslov method for propagating hydrodynamic waves in non-uniform currents. The problem of the influence of
large-scale oceanic motions on short internal wave propagation is considered
in their paper. This approach can also be generalized for describing waves in
water near the caustic both in the presence of non-uniform currents and with
uneven bottom relief.
Thus, the properties of the wave medium will be assumed to change along
the Ox axis and they are not dependent on time t. It follows from general
kinematic theory (see Chap. 1) that the wave vector component, directed
along the Oy axis, is constant, i.e. ky = kyo· The frequency w is persevered
along the ray as well:
w = F(kx, kyo, x) = const.
(5.48)
According to (1.24), the trajectory of the wave packet propagation can be
parametrically presented in the following form:
X
f c~x 1 dx = t - to ;
Xo
t
y = Yo + f Cgy dt ,
0
(5.49)
where the initial data are prescribed as xlt=to = xo; Yit=to = Yoi kxlt=to =
kxoi kylt=to =kyo· The wave amplitude at the initial moment a(x, y, t)it=to
= a0 is also considered to be known.
5 Wave Evolution in Non-uniform Currents in Deep Water
calculations. For this purpose a modified traditional scheme of the WKB approximation can be applied. In order to do so the Maslov method (Kravtsov,
1968; Maslov & Fedoryuk, 1976), describing the uniform wave field asymptotics in the entire space can be used. The Maslov method is often used in
obtaining a solution of the Helmholtz equation and it can be briefly described
as follows.
As mentioned above, the application of the geometrical optics method
leads to an amplitude value singularity in the caustic. However, when the
wave field is described by the integral representation the wave amplitude in
the caustic is finite in value. The Fourier integral can be used as the representation. Integration is with respect to the wave vector component k,
in which the direction the wave medium properties are changed (for example, along the Ox axis). In other words, one proceeds from the spatial coordinates { x1 = x, x2 = y} to the mixed spatial-momentum representation
{ k1 = kx, x 2 = y}. If the medium properties are changed in two directions
simultaneously, it is necessary to proceed from the spatial to the momentum
coordinates, i.e., to find a solution as the Fourier double integral.
However, the formal application of this method for waves in non-uniform
currents would be wrong, if the waves were described by a more complicated
equation than the Helmholtz equation (5.45), for which the aforementioned
asymptotic methods were initially developed. In this case the wave action
density is preserved rather than the energy. Voronovich and Goncharov (1982)
proposed the idea of generalizing the Maslov method for propagating hydrodynamic waves in non-uniform currents. The problem of the influence of
large-scale oceanic motions on short internal wave propagation is considered
in their paper. This approach can also be generalized for describing waves in
water near the caustic both in the presence of non-uniform currents and with
uneven bottom relief.
Thus, the properties of the wave medium will be assumed to change along
the Ox axis and they are not dependent on time t. It follows from general
kinematic theory (see Chap. 1) that the wave vector component, directed
along the Oy axis, is constant, i.e. ky = kyo· The frequency w is persevered
along the ray as well:
w = F(kx, kyo, x) = const.
(5.48)
According to (1.24), the trajectory of the wave packet propagation can be
parametrically presented in the following form:
X
f c~x 1 dx = t - to ;
Xo
t
y = Yo + f Cgy dt ,
0
(5.49)
where the initial data are prescribed as xlt=to = xo; Yit=to = Yoi kxlt=to =
kxoi kylt=to =kyo· The wave amplitude at the initial moment a(x, y, t)it=to
= a0 is also considered to be known.