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F(au/axe) = that the quantity f 1C12(Ia+(C)12 + la-(C)12) dC = and Parseval's theorem to show 2 f Iut(t,x)12 + IVXu(t,x)12 dx is independent of time for the solutions of the wave equation. The formula for a± are potentially singular at C = 0. The energy for the wave equation is expressed in terms of the pair of functions ICIa±(C). They are given by nonsingular expressions in terms of ICI f and 1. Simple Examples of Propagation 22 There are conservations of all orders. 4) ue(0, x) = y(x) eixl/E 'y E n Hs(Rd) .

Note that as -* oo, the group velocities approach ±1. High frequencies will propagate at speeds nearly equal to one. In particular they travel at the same speed. High frequency signals stay together better than low frequency signals. Since singularities of solutions are made of only the high frequencies (modifying the Fourier transform of the data on a compact set modifies the solution by a smooth term), one expects singularities to propagate at speeds ±1. 5. Once this is known for the fundamental solution, it follows for all.

The corresponding rays have velocities which differ by 0(e) so the rays remain close for times small compared with 1/c. 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(110The example reveals several scales of time. For times t << e, u and its gradient are well approximated by their initial values. For times e t << 1 u ei(X-t)/ea(0, x). The solution begins to oscillate in time.

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