1.2 Geometric Optics Approximation
15
The direction of wave propagation can be considered to be normal to the
wave surface in each small area.
The notion of rays (i.e. lines with tangents to them coinciding at every
point with the direction of wave propagation) is now introduced. 1 In geometric optics a ray is considered as wave propagation, digressing from their wave
nature. The geometric optics approximation corresponds to a limited case of
the small parameter c (here c = max{(M1k)- 1 , (M2k)- 1 , (Tw)- 1 } ).
The principal equations of geometric optics, describing the ray propagation, are now presented. Let the value ry(r, t) be the free surface deviation
from equilibrium. Thus, the value ' fJ for a plane monochromatic wave can be
written as:
' fJ = aei(kr-wt) = aei..P .
(1.16)
In the case of a wave different from the plane one, but with geometric
optics being applicable, the amplitude a is a function of the coordinates and
time a= a(r, t). The phase is written in a more complicated form than by
(1.16). However, it is essential for the phase to be of sufficiently large value
'¢ ~ 1 due to its change of 2n within the wavelength.
The expression (1.16) describes local sinusoidal waves. In small space areas
and a short time interval the phase '¢ can be expanded in the series:
(1.17)
Thus, the phase '¢ is connected with the local wave vector k and the local
frequency w:
8'¢
k = 8 r =grad('¢) ;
8'¢
w=-at.
It follows from the ratio (1.18) that:
rot(k) = 0,
(1.18)
(1.19)
(1.20)
i.e. the field of local wave vectors is non-vortical. This can be obtained from
(1.19):
8k
8t + grad(w) = 0,
(1.21)
which is a kinematic equation of wave density conservation (Phillips, 1980).
1 This determination corresponds to the case of wave propagation in isotropic media (Kravtsov & Or lov, 1980). The surface gravity waves in non-uniform currents
are attributed to dispersion waves in non-isotropic media. In this case the ray
will be more precisely defined below.