1.4. Fourier synthesis and rectilinear propagation 21 Fourier transformation with respect to the x variables yields ∂2ˆ(t, t ξ) + |ξ|2 ˆ(t, ξ) = 0 , ˆ(0,ξ) = ˆ, ∂tˆ(0,ξ) = ˆ . Solve the ordinary differential equations in t to find ˆ(t, ξ) = ˆ(ξ) cos t|ξ| + ˆ(ξ) sin t|ξ| |ξ| . For t = 0 the function ξ → sin(t|ξ|)/|ξ| is a real analytic function on Rd with derivatives uniformly bounded on bounded time intervals. Write cos t|ξ| = eit|ξ| + e−t|ξ| 2 , sin t|ξ| = eit|ξ| − e−t|ξ| 2i , to find (1.4.2) ˆ(t, ξ) eixξ = a+(ξ) ei(xξ−t|ξ|) − a−(ξ) ei(xξ+t|ξ|) , with (1.4.3) a+ := 1 2 ˆ + ˆ i|ξ| , a− := 1 2 ˆ − ˆ i|ξ| . For each ξ, the right-hand side of (1.4.2) is a linear combination of the plane wave solutions of the wave equation ei(xξ+tτ(ξ)) with dispersion relation τ = ∓|ξ| and amplitude a±(ξ). The group velocities associated to a± are v = −∇ξτ = −∇ξ(∓|ξ|) = ± ξ |ξ| . The solution is the sum of two terms, u±(t, x) := 1 (2π)d/2 a±(ξ) ei(xξ∓t|ξ|) dξ . The conserved energy for the spring equation satisfied by ˆ(t, ξ) shows that 1 2 |ˆt(t, ξ)|2 + |ξ|2|ˆ(t, ξ)|2 = |ξ|2 ( |a+(ξ)|2 + |a−(ξ)|2 ) = independent of t . Integrate dξ and use F ( ∂u/∂xj ) = iξj ˆ, and Parseval’s theorem to show that the quantity |ξ|2 ( |a+(ξ)|2 + |a−(ξ)|2 ) dξ = 1 2 |ut(t, x)|2 + |∇xu(t, x)|2 dx is independent of time for the solutions of the wave equation. The formula for a± are potentially singular at ξ = 0. The energy for the wave equation is expressed in terms of the pair of functions |ξ|a±(ξ). They are given by nonsingular expressions in terms of |ξ| ˆ and ˆ.

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