16
1 General Problem Formulation
Free waves can exist in the wave medium, not with the arbitrary frequency w and wave vector k, but those meeting some specific conditions.
In this case the frequency is a function of the wave vector w = F(k). The
function type is dependent on the wave motion type under consideration and
the corresponding force balance. However, in the case of a non-uniform and
non-stationary medium the frequency w depends not only on the wave vector k, but also on the coordinate r and time t. In the case of slow medium
parameter variation the dispersion ratio is local and can be written in the
form (Kravtsov & Orlov, 1980):
w = F(k, r, t) , k = k(r, t) .
(1.22)
Using the equations (1.18) and (1.19), the local dispersion ratio can be
rewritten in the form:
(1.23)
However, the phase equation (1.23) is significantly different from the dispersion ratio (1.22) in its content. It presents, not a simple algebraic correlation between the frequency and wave vector ratio, but a differential equation
in partial derivatives relative to the unknown function 1/J.
The equation (1.23) results in a remarkable analogy between geometric
optics and the mechanics of a material particle. The phase equation (1.23)
has the form of the Hamilton-Jacobi equation (Landau& Lifshits, 1973). It
is solved in mechanics relative to the particle action D, connected with the
momentum of the particle P and the Hamiltonian function 1£:
P = grad(D),
'1J = _ 8D
n
at .
By comparing these formulas with the ratios (1.18) and (1.19), it can be
seen that the action of the material particle D in mechanics plays the role
of the phase 1/J in geometric optics. The particle momentum P is analogous
to the wave vector k, and the Hamiltonian function 1l plays the role of the
frequency w (Landau&Lifshits, 1973). Thus, there is an analogy between
the behaviour of the material particle and the wave packet (i.e. the wave
representing a superposition of monochromatic waves with frequencies within
some small range and occupying a finite space area). The particle momentum
corresponds to the wave vector, and the particle energy is similar to the wave
packet frequency.
The characteristics of the (1.23) are determined by the system of ordinary
differential equations:
dr
8F
dk
dt
8k ; dt
8F
dw
8F
-or ; dt 8t.
(1.24)
1 General Problem Formulation
Free waves can exist in the wave medium, not with the arbitrary frequency w and wave vector k, but those meeting some specific conditions.
In this case the frequency is a function of the wave vector w = F(k). The
function type is dependent on the wave motion type under consideration and
the corresponding force balance. However, in the case of a non-uniform and
non-stationary medium the frequency w depends not only on the wave vector k, but also on the coordinate r and time t. In the case of slow medium
parameter variation the dispersion ratio is local and can be written in the
form (Kravtsov & Orlov, 1980):
w = F(k, r, t) , k = k(r, t) .
(1.22)
Using the equations (1.18) and (1.19), the local dispersion ratio can be
rewritten in the form:
(1.23)
However, the phase equation (1.23) is significantly different from the dispersion ratio (1.22) in its content. It presents, not a simple algebraic correlation between the frequency and wave vector ratio, but a differential equation
in partial derivatives relative to the unknown function 1/J.
The equation (1.23) results in a remarkable analogy between geometric
optics and the mechanics of a material particle. The phase equation (1.23)
has the form of the Hamilton-Jacobi equation (Landau& Lifshits, 1973). It
is solved in mechanics relative to the particle action D, connected with the
momentum of the particle P and the Hamiltonian function 1£:
P = grad(D),
'1J = _ 8D
n
at .
By comparing these formulas with the ratios (1.18) and (1.19), it can be
seen that the action of the material particle D in mechanics plays the role
of the phase 1/J in geometric optics. The particle momentum P is analogous
to the wave vector k, and the Hamiltonian function 1l plays the role of the
frequency w (Landau&Lifshits, 1973). Thus, there is an analogy between
the behaviour of the material particle and the wave packet (i.e. the wave
representing a superposition of monochromatic waves with frequencies within
some small range and occupying a finite space area). The particle momentum
corresponds to the wave vector, and the particle energy is similar to the wave
packet frequency.
The characteristics of the (1.23) are determined by the system of ordinary
differential equations:
dr
8F
dk
dt
8k ; dt
8F
dw
8F
-or ; dt 8t.
(1.24)