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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) .