Geoscience Reference
In-Depth Information
d
X
d
t
k
x
k
x
+
k
y
=
c
,
(5.11)
d
Y
d
t
k
y
k
x
+
k
y
=
c
,
(5.12)
k
x
+
k
y
,
−
∂
d
k
x
d
t
c
=
(5.13)
∂
x
k
x
+
k
y
,
d
k
y
d
t
−
∂
c
=
(5.14)
∂
y
where
x
and
y
are the coordinates of ray points,
k
x
and
k
y
are components of
the wave vector and
t
represents time. The ray equations written in this form, per-
mit not only to easily calculate the course of the wave ray, but also the evolution of
the wave front. For computing the ray evolution the set of equations (5.11)-(5.14)
must be supplemented by initial conditions consisting in determination of the initial
coordinates and direction. It is not difficult to note that in the case of a fixed basin
depth (
c
= const) the rays will be straight lines.
Figure 5.7 presents two examples of the computation of ray behaviour, performed
with the aid of formulae (5.11)-(5.14). The model bottom relief in the first case
(Fig. 5.7a) imitates a mountain ridge in the middle of the ocean. Part of the rays
emitted by a point-like source turn out to be captured, these rays further propagate
(a)
(b)
Fig. 5.7 Examples of influence of bottom relief on the course of wave rays emitted by a pointlike
source: capture of rays by an underwater ridge (a); refraction and capture of waves in the shelf
zone (b)