1.4. Fourier synthesis and rectilinear propagation 25
The approximation retains some accuracy so long as t = o(1/).
The approximation has the following geometric interpretation. One has
a superposition of plane waves
ei(xξ−t|ξ|)
with ξ = (1/, 0,..., 0) + O(1).
Replacing ξ by (1/, 0,..., 0) and |ξ| by 1/ in the plane waves yields the
approximation (1.4.6).
The wave vectors, ξ, make an angle O( ) with e1. The corresponding
rays have velocities which differ by O( ) so the rays remain close for times
small compared with 1/. For longer times the fact that the group velocities
are not parallel is important. The wave begins to spread out. Parallel group
velocities are a reasonable approximation for times t = o(1/).
The example reveals several scales of time. For times t , u and
its gradient are well approximated by their initial values. For times
t 1 u
ei(x−t)/a(0,x).
The solution begins to oscillate in time. For
t = O(1) the approximation u A(t, x)
ei(x−t)/
is appropriate. For times
t = O(1/) the approximation ceases to be accurate. Refined approximations
valid on this longer time scale are called diffractive geometric optics. The
reader is referred to [Donnat, Joly, etiver, and Rauch, 1995–1996] for an
introduction in the spirit of Chapters 7–8.
It is typical of the approximations of geometric optics that
(
uapprox uexact
)
= uapprox = O(1)
is not small. The error uapprox uexact = O( ) is smaller by a factor of .
The residual uapprox oscillates on the scale , and after applying
−1
it is
smaller by a factor .
The analysis just performed can be carried out without fundamental
change for initial oscillations with nonlinear phase. An excellent descrip-
tion, including the phase shift on crossing a focal point, can be found in
[H¨ ormander 1983, §12.2].
Next the approximation is pushed to higher accuracy with the result
that the residuals can be reduced to O(
N
) for any N. Taylor expansion to
higher order yields
(1.4.10) |e1 + η| = 1 + η1 +
|α|≥2
cαηα
, |η| 1,
so
(|e1 + ζ| 1
)
/ ζ1 +
|α|≥2
|α|−1

ζα,
eit(|e1+ζ|−1)/

eitζ1
e

|α|≥2
it|α|−1 cαζα

eitζ1
1 +
j≥1
j
hj(t, ζ) .
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