192
5 Wave Evolution in Non-uniform Currents in Deep Water
y
Caustic
Shadow zone
X
Fig. 5.17. Ray turning in caustic area
Taking a quantum-mechanical analogy (Kravtsov & Orlov, 1980; Landau
& Lifshits, 1974a), it can be said that (5.45) is also the linear Schrodinger
equation describing the stationary wave function for a particle with mass m
and energy E in a force field with potential U(z), where k0 is the value
J2mE/n; n(z) is the value 1- U(z)/E, so that k5n 2 = 2m(E- U)/n 2 (n is
the Planck constant).
The turning points z = z* are analogues to the caustic, with the coefficient n(z) becoming zero in problems where a wave field is described by the
ordinary differential equation (5.45). The term "turning point" corresponds
to the case n = 0, when the particle kinetic energy is compared to the potential energy and the particle changes its direction of motion. At the same
time there is also a turning point in the ray trajectory in coordinate space
(see Fig. 5.17).
Smooth caustic surfaces without any special points are called ordinary.
In the area of an ordinary caustic, two rays intersect, touching this caustic
(see Fig. 5.17). The peculiarities in the caustic (points of sharpening, loops,
etc.) are accompanied by a great number of rays intersecting at one point.
Generally, the classification of different caustic types is based on the theory
of differentiated reflection peculiarities, also called the "theory of catastrophes" (Arnold, 1989). An ordinary caustic is called "a fold" according to the
terminology of this theory.
The important role of caustics in wave problems is determined by characterizing the ray family, permitting it to form the entire space field pattern.
The family of the rays can be restored with the help of ray directions at the
caustic surface. The areas where rays cannot penetrate are called the "caustic
shadow zone". A zero field corresponds to the caustic shadow zone in the geo-
5 Wave Evolution in Non-uniform Currents in Deep Water
y
Caustic
Shadow zone
X
Fig. 5.17. Ray turning in caustic area
Taking a quantum-mechanical analogy (Kravtsov & Orlov, 1980; Landau
& Lifshits, 1974a), it can be said that (5.45) is also the linear Schrodinger
equation describing the stationary wave function for a particle with mass m
and energy E in a force field with potential U(z), where k0 is the value
J2mE/n; n(z) is the value 1- U(z)/E, so that k5n 2 = 2m(E- U)/n 2 (n is
the Planck constant).
The turning points z = z* are analogues to the caustic, with the coefficient n(z) becoming zero in problems where a wave field is described by the
ordinary differential equation (5.45). The term "turning point" corresponds
to the case n = 0, when the particle kinetic energy is compared to the potential energy and the particle changes its direction of motion. At the same
time there is also a turning point in the ray trajectory in coordinate space
(see Fig. 5.17).
Smooth caustic surfaces without any special points are called ordinary.
In the area of an ordinary caustic, two rays intersect, touching this caustic
(see Fig. 5.17). The peculiarities in the caustic (points of sharpening, loops,
etc.) are accompanied by a great number of rays intersecting at one point.
Generally, the classification of different caustic types is based on the theory
of differentiated reflection peculiarities, also called the "theory of catastrophes" (Arnold, 1989). An ordinary caustic is called "a fold" according to the
terminology of this theory.
The important role of caustics in wave problems is determined by characterizing the ray family, permitting it to form the entire space field pattern.
The family of the rays can be restored with the help of ray directions at the
caustic surface. The areas where rays cannot penetrate are called the "caustic
shadow zone". A zero field corresponds to the caustic shadow zone in the geo-