Frequency-dependent damping in the linear wave equation

TL;DR

We address damping in the linear wave equation that acts only on selected spatial frequencies via a bounded monotone operator $P$, i.e. $\partial_{tt} u - \Delta u + P[\partial_t u] = 0$. We prove well-posedness using maximal monotone operator theory, establish an energy-dissipation balance $E(u(t),v(t)) + \int_0^t \langle P[v(s)], v(s)\rangle ds = E(u_0,v_0)$, and analyze explicit 1D solutions when $P$ is a Fourier multiplier; the work also shows a precise splitting into dissipative and conservative components in special cases. The results illuminate how frequency-selective damping shapes long-time behavior of linear waves and provide a foundation for extending to nonlinear damping and design of spectral filters in wave propagation models.

Abstract

We propose a model for frequency-dependent damping in the linear wave equation. After proving well-posedness of the problem, we study qualitative properties of the energy. In the one-dimensional case, we provide an explicit analysis for special choices of the damping operator. Finally, we show, in special cases, that solutions split into a dissipative and a conservative part.

Figures (1)

• Figure 1: Numerical simulation of the solution to the problem \ref{['eq:PDE 1d']} with $L = 1$, $u_0$ built on modes with frequencies ranging from $0$ to $20$, $v_0 = 0$, and $\phi$ such that $\widehat{\phi}_k = 1$ for $|k| < 3$ and $\widehat{\phi}_k = 0$ for $|k| \geq 3$. The figure shows on top left the initial condition $u_0$ in solid black and its projection $Q[u_0]$ on the null space of $P$ in dashed black. The other frames show the solution $u(t,\cdot)$ at different times. The solution converges exponentially fast to the solution built only on the modes with low frequencies.

Theorems & Definitions (23)

• Remark 2.1
• Example 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Theorem 2.6
• proof
• Remark 2.7