1.2 Geometric Optics Approximation
17
The equations (1.24) are a system of Hamiltonian equations. The solution
{r(t), t} of (1.24) determines the spatial-temporal rays in three-dimensional
space {x,y,t}. The spatial rays r = r(t) are projections of spatial-temporal
rays onto the coordinate space r = { x, y}.
It follows from (1.24) that a wave packet moves at the group velocity:
(1.25)
The second equation (1.24) characterizes the change of the wave vector
along the ray. The third equation (1.24) describes the frequency variation.
The frequency remains constant along the ray: w = constant in a stationary
medium, with a dispersion ratio (1.22) without being explicitly dependent on
time.
An expression for the wave phase ' ljJ along the characteristics can be obtained similarly. Using the determination of the wave action D as an integral
by the Lagrangian function L, it can be written as follows:
t
t
D = Do + J L dt = Do + J ( P ~; - 1i) dt . (1.26)
to
to
Thus, the following wave phase expression is obtained:
t
'¢ = ' l /Jo + j (kG g - w) dt ,
(1.27)
to
where '¢0 is the initial phase value.
The second term in the expression (1.27) is reduced to zero in a medium
without dispersion. In this case the group velocity C g coincides with the
phase velocity C = kw/k 2 . Thus, the phase is the preserved value ' ljJ =
'¢0 in spatial-temporal rays. In the dispersion medium the so-called speed
group delay, determined by the second item in the expression (1.27), arises
(Kravtsov & Orlov, 1980). The group speed delay means a displacement of
the wave packet propagation velocity in comparison with the phase velocity.
All these facts remain in force in the medium moving with speed V, as
long as it changes slowly enough. The velocity V can be pointed out in the
equations as follows. Let the value r be a spatial vector in the immovable
coordinate system, and the value r 1 be a proper local vector in the coordinate
system, moving alongside the medium. Then r 1 = r- Vt.
As a result of the transition to the new variable r 1 , the Hamilton-Jacobi
equation determining the phase (1.23) is written in the form:
17
The equations (1.24) are a system of Hamiltonian equations. The solution
{r(t), t} of (1.24) determines the spatial-temporal rays in three-dimensional
space {x,y,t}. The spatial rays r = r(t) are projections of spatial-temporal
rays onto the coordinate space r = { x, y}.
It follows from (1.24) that a wave packet moves at the group velocity:
(1.25)
The second equation (1.24) characterizes the change of the wave vector
along the ray. The third equation (1.24) describes the frequency variation.
The frequency remains constant along the ray: w = constant in a stationary
medium, with a dispersion ratio (1.22) without being explicitly dependent on
time.
An expression for the wave phase ' ljJ along the characteristics can be obtained similarly. Using the determination of the wave action D as an integral
by the Lagrangian function L, it can be written as follows:
t
t
D = Do + J L dt = Do + J ( P ~; - 1i) dt . (1.26)
to
to
Thus, the following wave phase expression is obtained:
t
'¢ = ' l /Jo + j (kG g - w) dt ,
(1.27)
to
where '¢0 is the initial phase value.
The second term in the expression (1.27) is reduced to zero in a medium
without dispersion. In this case the group velocity C g coincides with the
phase velocity C = kw/k 2 . Thus, the phase is the preserved value ' ljJ =
'¢0 in spatial-temporal rays. In the dispersion medium the so-called speed
group delay, determined by the second item in the expression (1.27), arises
(Kravtsov & Orlov, 1980). The group speed delay means a displacement of
the wave packet propagation velocity in comparison with the phase velocity.
All these facts remain in force in the medium moving with speed V, as
long as it changes slowly enough. The velocity V can be pointed out in the
equations as follows. Let the value r be a spatial vector in the immovable
coordinate system, and the value r 1 be a proper local vector in the coordinate
system, moving alongside the medium. Then r 1 = r- Vt.
As a result of the transition to the new variable r 1 , the Hamilton-Jacobi
equation determining the phase (1.23) is written in the form: