22
1 General Problem Formulation
The equation r = r((, (, r) determines a set of rays generated by the prescribed field distribution in the initial surface r(To) = r 0 ((,(). The equation
for a set of rays describes a relation between the ray coordinates and the
Cartesian coordinates. If the Jacobian J1 = 8(x, y, z)/8((, (, r) is not equal
to zero in the considered area, then the equation r = r((, (, T) can be solved
unambiguously relative to the ray coordinates corresponding to the point of
observation ( = ((r), ( = ((r), T = r(r).
The results obtained in this chapter give the solution of wave propagation
in a water surface with a horizontal non-uniform current and uneven bottom
as follows:
( t) -
J-1/2 J.-1/2 i.Y
TJ r, - ao 1
2
e ,
(1.46)
where the wave phase ' 1/J is determined by the initial conditions according to
(1.26): '1/J(r, t) = '1/J(ro) + J~(k Cg- w) dt.
Unlike the classical case (Kravtsov & Orlov, 1980) an additional multiplier J2 = a0 j a appears in the expression (1.46) due to the influence of the
spatial non-uniform currents, so long as the equation of wave action density
conservation (1.41) is solved instead of that for the energy.
An important result of the solution (1.46) is that the following equality
holds along the characteristics (Kravtsov & Orlov, 1980):
ICgAI dl = const ,
(1.47)
where dl is the distance between two infinitely close projections of the characteristics towards the coordinate space { x, y }. It follows from (1.24) that the
ratio (1.47) establishes the preservation law of wave action flow along a ray
tube. Another simple consequence of (1.24) and (1.47) should be noted. If
the medium properties are not dependent on time t, then the frequency w is
preserved. Additionally the value of the wave vector component kx is also preserved along the characteristics in the spatial-cylindrical case (i.e. the wave
medi urn properties depend only on one coordinate y). These characteristics
are parallel lines (see Fig. 1.1). The ratio (1.47) can be written in the simple
form: CgyA = const. These kinds of ratios are used for solving a wide range
of problems. For example, they describe wave propagation in shallow water
with a depth varing only along one direction, i.e. at parallel isobaths or in
the presence of horizontal-shear currents. A number of such problems will be
considered below.
The results presented in this chapter allow us to consider in general wave
propagation with slow temporal and weak horizontal non-uniformity in the
average medium state of the ocean.
The sphere of application of this theory should be made clear. The aforementioned methods describing waves in water are based on the assumption
of waves being locally plane. However, this assumption is not always fulfilled.
Sometimes there are situations in which changes, small in comparison with
the wavelength, are accumulated. This may lead to the phenomenon where
1 General Problem Formulation
The equation r = r((, (, r) determines a set of rays generated by the prescribed field distribution in the initial surface r(To) = r 0 ((,(). The equation
for a set of rays describes a relation between the ray coordinates and the
Cartesian coordinates. If the Jacobian J1 = 8(x, y, z)/8((, (, r) is not equal
to zero in the considered area, then the equation r = r((, (, T) can be solved
unambiguously relative to the ray coordinates corresponding to the point of
observation ( = ((r), ( = ((r), T = r(r).
The results obtained in this chapter give the solution of wave propagation
in a water surface with a horizontal non-uniform current and uneven bottom
as follows:
( t) -
J-1/2 J.-1/2 i.Y
TJ r, - ao 1
2
e ,
(1.46)
where the wave phase ' 1/J is determined by the initial conditions according to
(1.26): '1/J(r, t) = '1/J(ro) + J~(k Cg- w) dt.
Unlike the classical case (Kravtsov & Orlov, 1980) an additional multiplier J2 = a0 j a appears in the expression (1.46) due to the influence of the
spatial non-uniform currents, so long as the equation of wave action density
conservation (1.41) is solved instead of that for the energy.
An important result of the solution (1.46) is that the following equality
holds along the characteristics (Kravtsov & Orlov, 1980):
ICgAI dl = const ,
(1.47)
where dl is the distance between two infinitely close projections of the characteristics towards the coordinate space { x, y }. It follows from (1.24) that the
ratio (1.47) establishes the preservation law of wave action flow along a ray
tube. Another simple consequence of (1.24) and (1.47) should be noted. If
the medium properties are not dependent on time t, then the frequency w is
preserved. Additionally the value of the wave vector component kx is also preserved along the characteristics in the spatial-cylindrical case (i.e. the wave
medi urn properties depend only on one coordinate y). These characteristics
are parallel lines (see Fig. 1.1). The ratio (1.47) can be written in the simple
form: CgyA = const. These kinds of ratios are used for solving a wide range
of problems. For example, they describe wave propagation in shallow water
with a depth varing only along one direction, i.e. at parallel isobaths or in
the presence of horizontal-shear currents. A number of such problems will be
considered below.
The results presented in this chapter allow us to consider in general wave
propagation with slow temporal and weak horizontal non-uniformity in the
average medium state of the ocean.
The sphere of application of this theory should be made clear. The aforementioned methods describing waves in water are based on the assumption
of waves being locally plane. However, this assumption is not always fulfilled.
Sometimes there are situations in which changes, small in comparison with
the wavelength, are accumulated. This may lead to the phenomenon where