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, M´ 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

cα

ζα,

eit(|e1+ζ|−1)/

∼

eitζ1

e

∑

|α|≥2

it|α|−1 cαζα

∼

eitζ1

1 +

j≥1

j

hj(t, ζ) .