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Localizing acoustic and electromagnetic waves in space and time

Roland Griesmaier, Soumen Senapati

Abstract

We study time-dependent acoustic and electromagnetic waves governed by the scalar wave equation or Maxwell's equations in a bounded three-dimensional domain. We establish the existence of time-dependent boundary excitations that can be prescribed on any open subset of the boundary of the domain such that the associated waves are strongly localized in space in the sense that they possess arbitrarily large norms in a given subdomain and on a given time-interval, while remaining arbitrarily small in any other given subdomain for all times. Similarly, we also show the existence of boundary data such that the associated waves are strongly localized in time in the sense that they possess arbitrarily large norms in a given subdomain and on a given time-interval, while remaining arbitrarily small on the same subdomain but on any other prescribed time-interval. In case that we have access to the possibly inhomogeneous coefficients in the wave equation or in the Maxwell system, we also give explicit constructions to obtain boundary data that generate these localized waves, and we comment on possible applications.

Localizing acoustic and electromagnetic waves in space and time

Abstract

We study time-dependent acoustic and electromagnetic waves governed by the scalar wave equation or Maxwell's equations in a bounded three-dimensional domain. We establish the existence of time-dependent boundary excitations that can be prescribed on any open subset of the boundary of the domain such that the associated waves are strongly localized in space in the sense that they possess arbitrarily large norms in a given subdomain and on a given time-interval, while remaining arbitrarily small in any other given subdomain for all times. Similarly, we also show the existence of boundary data such that the associated waves are strongly localized in time in the sense that they possess arbitrarily large norms in a given subdomain and on a given time-interval, while remaining arbitrarily small on the same subdomain but on any other prescribed time-interval. In case that we have access to the possibly inhomogeneous coefficients in the wave equation or in the Maxwell system, we also give explicit constructions to obtain boundary data that generate these localized waves, and we comment on possible applications.
Paper Structure (8 sections, 16 theorems, 119 equations)

This paper contains 8 sections, 16 theorems, 119 equations.

Table of Contents

  1. Introduction.
  2. The mathematical setting.
  3. Localized solutions for the acoustic wave equation.
  4. Localization in space
  5. Localization in time
  6. Localized solutions for the Maxwell equations.
  7. Localization in space
  8. Localization in time

Key Result

Proposition 2.1

For any $f\in L^2(\Gamma\times(0,T))$, there exists a unique weak solution $u_f$ of eq:IBVP_acoustic. This solution satisfies $u_f \in C ([0,T]; L^2(\Omega) )$, and it depends continuously on the data.

Theorems & Definitions (33)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 23 more