Damped wave equation ($u_{tt}+\alpha u_t=c^2{u}_{xx}$)

A wave equation with a linear damping term and positive wave speed $c$ separates into spatial normal modes and damped harmonic oscillators. For a spatial eigenvalue $k^2$, the time factor obeys $T^{\prime \prime }+\alpha T^{\prime }+c^2{k}^{2}T=0$. Its characteristic roots are $-\alpha /2\pm \sqrt{{\alpha }^2/4-c^2{k}^2}$. Thus underdamping requires $|\alpha |<2c|k|$, critical damping gives a repeated root and the second solution $t{e}^{-\alpha t/2}$, and overdamping gives two real exponential rates. Negative $\alpha$ amplifies rather than damps the motion.

For the operator $c^{-2}{\partial }_t^2+2\alpha {\partial }_t-{\partial }_x^2$ with $c>0$, the spatial Fourier mode $k$ of its causal Green function solvesSet $\gamma =\alpha c^2$ and $d_k^2={\gamma }^2-c^2{k}^2$. Causality, continuity at zero and the derivative jump $G_k^{\prime }(0^{+})=c^2$ giveImaginary $d_k$ gives a sine; when $d_k=0$, the quotient has limit $t$. These cases are one continuous formula, rather than separate arbitrary normalizations.

The spatially uniform mode isOnly for positive $\alpha$ does it tend to $1/(2\alpha )$. With zero damping it grows linearly, and negative $\alpha$ produces amplification. A stationary zero-mode limit therefore requires a sign hypothesis even though the causal formula itself covers all real $\alpha$.

At small wavenumber compared with $\alpha c$, the slow decay rate of the damped wave operator is $k^2/(2\alpha )$ and the other root decays on time $1/(2\alpha c^2)$. Long-time, long-wavelength motion therefore approaches heat diffusion with the displayed diffusivity. The original hyperbolic equation nevertheless has finite signal speed $c$, unlike the parabolic diffusion approximation.