The modulated Fourier expansion for waves propagating through time-modulated media

TL;DR

The paper introduces a modulated Fourier expansion (MFE) for the time-modulated acoustic wave equation to overcome difficulties of time-harmonic formulations in nonreciprocal media. By deriving a coupled, linear time-invariant system for slowly varying coefficient functions $z^K_k$, it proves well-posedness in the Laplace domain, establishes decay bounds for the expansion coefficients, and provides rigorous error estimates for the truncated expansion. It further shows that time discretization via convolution quadrature yields stable integrators with step sizes independent of the modulation period $\varepsilon$ for small modulation amplitude $\rho$, and derives energy estimates and quasi-conservation properties for short times. The approach reduces the original problem to solving a single spatial operator per time step, enabling efficient simulations and preconditioning, with numerical experiments in 1D validating decay of the coefficients and the practical performance of the method.

Abstract

Controlling waves by actively changing the material parameters of a medium enables the development of new acoustic and electrical devices. Modulating the material breaks classical properties like reciprocity and the conservation of energy, which complicates the mathematical analysis. Without a limiting amplitude principle, time-harmonic formulations are generally inapplicable. The present manuscript develops an alternative tool for the time-modulated acoustic wave equation, that is based on a modulated Fourier expansion (MFE). The solution is characterized by multiple smoothly varying coefficient functions, which solve a coupled system of evolutionary partial differential equations with temporally constant coefficients. For small-amplitude fast-time modulations, this system of evolutionary partial differential equations is shown to possess a smoothly varying solution, which characterizes the exact solution up to a small defect. Discretization of the derived coupled system yields integrators that are stable and accurate when larger time steps are used, compared to those schemes that are applied to the time-modulated acoustic wave equation directly. Numerical experiments illustrate the theoretical results and the use of the approach.

Figures (5)

• Figure 1: Time convergence error for $\rho = 0.4$ and $\varepsilon = 0.04$, measured in the space time $L^2-$ norm in time and space. The solution is computed until the final time $T=5$ and the MFE is used with $K=3$. A finite difference solver is used for space discretization with $1000$ degrees of freedom.
• Figure 2: Visualization of $u$, with $500$ spatial degrees of freedom and $N=256$ time steps. The parameters were set to $\rho=0.4$ and $\varepsilon=0.04$.
• Figure 3: The absolute values of the coefficient functions $z_0,z_1$ and $z_2$ that were used to compute the approximation of Figure \ref{['fig:visualization']}.
• Figure 4: Spatio-temporal $L^2$-norm of the modulation functions $z_k$ for $k=0,\dots,K$ with $K=10$, for different values of $\varepsilon$ and $\rho$, with the regular right-hand side $f$ of \ref{['eq:source']}.
• Figure 5: The same plot as Figure \ref{['fig:z_K']}, but for the excitation $f$ described in Equation \ref{['eq:f-low-reg']}, with low spatial regularity.

Theorems & Definitions (44)

• Theorem 1: Modulated Fourier expansion
• Theorem 2: Modulated Fourier expansion for low spatially variable $\mu$
• Remark 2.1: Rapidly oscillating excitations
• Remark 2.2: Limitations of the approach
• Remark 2.3: Symmetry of the coefficients
• Remark 3.1: Connection to coupled harmonics
• Lemma 1: Coercivity of $a_K$
• proof
• Lemma 2
• Proposition 1