Fourier transform method for the wave equation

Fourier transformation in space turns $u_{tt}=c^2{u}_{xx}$ into the independent oscillators ${\hat{u}}_{tt}+c^2{k}^2\hat{u}=0$.

For zero initial displacement and unit delta initial velocity, the spatial Fourier transform of the wave equation gives the displayed multiplier. A compact cutoff function around zero frequency produces an ordinary smooth function after inversion. The remaining sine splits into two oscillatory integrals with phases $x\cdot \xi \pm ct|\xi |$ and symbols proportional to $(1-\chi )/|\xi |$ of order $-1$. Their stationary directions lie on $|x|=c|t|$, proving a light-cone bound for singular support. This does not claim vanishing of smooth tails inside the cone.

For real wave speed $c$ and $t\ge 0$, define the complex-frequency cosine through the entire series $C_t(z)={\sum }_{j\ge 0}(-1{)}^j(ct{)}^{2j}(z\cdot z{)}^j/(2j)!$. The expression is independent of the square-root branch even though the square root itself need not be entire. Writing $z=a+ib$ gives $|\mathrm{Im}\sqrt{z\cdot z}|\le |b|$, and hence $|C_t(z)|\le e^{|c|t|\mathrm{Im}z|}$. The Paley–Wiener–Schwartz theorem therefore makes its inverse Fourier transform a distribution supported in the radius-$|c|t$ ball. Multiplication by $C_t$ evolves initial displacement with zero initial velocity in the wave equation, giving finite propagation speed for compactly supported initial distributions.