<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\makeatletter
\@ifundefined{HCode}
{\documentclass[CRPHYS,Unicode,screen,biblatex,published]{cedram}
\addbibresource{crphys20250150.bib}
\newenvironment{noXML}{}{}
\def\tsup#1{$^{{#1}}$}
\def\tsub#1{$_{{#1}}$}
\def\sfrac#1#2{{#1}/{#1}}
\usepackage{amssymb}
\usepackage{upgreek}
\def\thead{\noalign{\relax}\hline}
\def\tbody{\noalign{\relax}}
\def\endthead{\noalign{\relax}\hline}
\def\tabnote#1{\vskip4pt\parbox{.90\linewidth}{#1}}
\RequirePackage{etoolbox}
\usepackage{amsmath}
\usepackage{dsfont}
\usepackage{marvosym}
\csdef{Seqnsplit}{\\}
\def\tbody{\noalign{\relax}\hline}
\def\jobid{crphys20250150}
%\graphicspath{{/tmp/\jobid_figs/web/}}
\graphicspath{{./figures/}}
\def\xsection{}
\newcounter{runlevel}
\let\MakeYrStrItalic\relax
\def\refinput#1{}
\def\0{\phantom{0}}
\def\sfrac#1#2{{#1}/{#2}}
\def\stfrac#1#2{({#1})/{#2}}
\def\stofrac#1#2{(({#1})/{#2})}
\def\back#1{}
\def\textemail#1{\href{mailto:#1}{#1}}
\def\botline{\\\hline}
\def\rmpi{\uppi}
\DOI{10.5802/crphys.242}
\datereceived{2024-11-25}
\daterevised{2025-02-12}
\dateaccepted{2025-02-24}
\ItHasTeXPublished
\makeatletter
\g@addto@macro{\UrlBreaks}{\UrlOrds}
\gappto{\UrlBreaks}{\UrlOrds}
\makeatother
}
{\documentclass[crphys]{article}
\def\CDRdoi{10.5802/crphys.242}
\let\newline\break
\def\Hfill{}
\usepackage[T1]{fontenc} 
\usepackage{marvosym}
\makeatletter
\def\sfrac#1#2{{#1}/{#1}}
\def\CDRsupplementaryTwotypes[#1]#2#3{}
}
\makeatother

\usepackage{upgreek}

\dateposted{2025-03-05}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Coupled dynamics of a wave and moving boundary}

\alttitle{Dynamique coupl\'{e}e d'une onde et d'une paroi mobile}

\author{\firstname{Guillaume} \lastname{Michel}\CDRorcid{0000-0003-3982-5838}}
\address{Sorbonne Universit\'{e}, CNRS, Institut Jean Le Rond d'Alembert, F-75005 Paris, France}
\email{guillaume.michel@sorbonne-universite.fr}

\begin{abstract}
I present the dynamics of waves trapped in a one-dimensional cavity
with a single wall that can move as a result of radiation pressure (and
possibly other external forces). Specifically, the classical wave
equation is considered and the phenomenology of this system is outlined
when the moving boundary achieves large displacements and velocities
similar to the one of the waves. Governing equations are derived in the
context of electromagnetism, and a spatial discretization that
conserves the total energy is proposed. I address cases  when the
dynamics of the boundary are prescribed or critically damped. Finally,
numerical simulations are performed to obtain qualitative results for
the conservative limit. 
\end{abstract}

\begin{altabstract} 
Je pr\'{e}sente la dynamique d'ondes dans une cavit\'{e}
unidimensionnelle dont une paroi peut se mettre en mouvement sous
l'effet, entre autres, de la pression de radiation. Plus
pr\'{e}cis\'{e}ment, l'\'{e}quation des ondes classiques est
coupl\'{e}e \`a une paroi subissant de grands d\'{e}placements et
pouvant se mouvoir \`a une vitesse comparable \`a celle des ondes. Les
\'{e}quations du probl\`{e}me sont obtenues dans le cadre de
l'\'{e}lectromagn\'{e}tisme puis discr\'{e}tis\'{e}es spatialement en
conservant l'\'{e}nergie totale. Quelques cas particuliers sont ensuite
trait\'{e}s, lorsque la position de la paroi est prescrite par un
op\'{e}rateur ext\'{e}rieur ou bien se r\'{e}duit \`a un oscillateur
dans son r\'{e}gime critique. Des simulations num\'{e}riques dans la
limite conservatives sont finalement discut\'{e}es.
\end{altabstract}

\keywords{\kwd{Wave} 
\kwd{Radiation pressure}
\kwd{Moving boundary}
\kwd{Nonlinear dynamics}
\kwd{Doppler effect}}

\altkeywords{\kwd{Ondes} 
\kwd{Pression de radiation} 
\kwd{Paroi mobile} 
\kwd{Dynamique non lin\'{e}aire} 
\kwd{Effet Doppler}}

\def\thanksname{Note}
\thanks{Guillaume Michel is the 2024 recipient of the Jacques
Herbrand/Fondation Mireille Cahn-Bunel prize of the Acad\'emie des
Sciences.}

\maketitle

\vspace*{-.5pc}

\end{noXML}

\def\eqref#1{(\ref{#1})}

\section{Introduction}

The Doppler effect constitutes a canonical example of an interaction
between a wave and a moving boundary: traveling solutions of the wave
equation present changes in both their energy and  frequency as they
reflect off a uniformly moving boundary~\cite{Goedeck_2012}. This
result has been widely generalized to non-uniform motions. For
instance, quasi-monochromatic waves of frequency $f_0$ reflecting off a
harmonically oscillating boundary at frequency $f_b$ acquire a spectrum
of the form $\{ f_0 + n f_b \}_{n \in \mathbb{Z}}$ that can
be explicitly computed~\cite{Censor_1973}. This holds for
electromagnetic waves and---after much controversy over the comparative
effects of the medium non-linearity~\cite{Censor_1988}---has also been
experimentally confirmed  for acoustic waves~\cite{Mujica_2009}. This
approach can be adapted for surface gravity waves, even though they are
not governed by the standard wave equation~\cite{Michel_2016}.

A natural extension of these works is to consider the cumulative
effects of such generalized Doppler shifts for waves trapped in a
cavity with oscillating walls. The one-dimensional wave equation in the
modulated domain $[0,L_0 + A \cos(2\rmpi f_b t)]$ with non-zero initial
conditions leads, in some range of parameters, to a localization and
exponential growth in wave energy~\cite{Dittrich_1994, Meplan_1996}.
This parametric instability is known as the dynamical Casimir effect in
quantum electrodynamics and can generate photons in a 
vacuum~\cite{Dodonov_2010}. This instability is also considered in the
construction of ultra-short pulses~\cite{Copper_1993} and contributes
to the interpretations of experiments on wave
turbulence~\cite{Michel_2016}. In practice, the motion of the
oscillating wall becomes increasingly difficult to sustain as the wave
energy continues to grow.

In the problems discussed above, the position of the moving boundary is
assumed to be prescribed regardless of the radiation pressure generated
by the waves. Taking this feedback into account enables a large class
of new phenomena related to thermalization and heat fluxes to be
investigated. Consider, for instance, the setup shown in 
Figure~\ref{fig:setup}({a}), in which a wave propagates in a
cavity whose right wall is connected to a spring. The energy of this
entire system is conserved, but does it  eventually thermalize? And how
long does it take? Similar questions arise if two of these moving
boundaries are now introduced (the left and right walls), and connected
to thermostats of different temperatures, see
Figure~\ref{fig:setup}({b}). Is there a mean heat flux in this
system? If yes, how does it scale with the temperature difference? Does
it depend on the initial wave energy?

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:setup}Two examples of one-dimensional wave fields coupled with
moving boundaries. ({a}) A model that allows energy conservation
and investigation into the characteristics of the stationary state.
({b}) A model of heat transfer mediated by radiation pressure
(i.e., caused neither by conduction or convection nor by emission or
absorption of radiation).}
\end{figure}

As a result of the tremendous potential applications of
optomechanics~\cite{RevModPhys.86.1391, Barzanjeh}, the modeling of
such systems, initially based on the classical wave equation, nowadays
involves quantized electromagnetic fields and boundary
motions~\cite{Dodonov_review}. The framework developed by
Law~\cite{Law_1995} is routinely used (see, e.g.,~\cite{Butera,
Nian_2021, PhysRevLett.122.030402} and references therein). It
crucially relies on both nonrelativistic and small-displacements
assumptions; in addition, most studies restrict to a few
electromagnetic modes (generally a single one). In contrast, I focus in
this manuscript on the opposite limit of cavities driven by radiation
pressure in which a large number of wave modes interact with a boundary
that undergoes large displacements and can move up to the speed of the
waves. I try to provide here a pedagogical introduction to the
governing equations and emerging properties as well as a simple
numerical scheme able to capture this complex dynamics. Classical
electromagnetism is used for that purpose, but the general picture is
expected to hold for mechanical waves as well.

The remainder of this manuscript is organized as follows. The radiation
pressure is derived from the wave equation in
Section~\ref{S:rad_press}. I introduce in Section~\ref{S:g_eq}  the
governing equations of the problem displayed in
Figure~\ref{fig:setup}({a}) (({b}) being an immediate
extension) and propose an energy-conserving discretization in 
Section~\ref{S:numeric}. I then discuss a few cases in which
theoretical predictions can be drawn in Section~\ref{S:simple_limits}
and investigate the fully conservative limit using numerical
simulations in Section~\ref{S:DNS}. 

\section{The radiation pressure and the wave equation}\label{S:rad_press}

The investigation of the coupling between a wave field and a moving
boundary requires an expression for the radiation pressure. There is a
simple way to derive its expression:  it is reported here as, in my
opinion, one of the most straightforward introduction to radiation
pressure that can be done in classroom. Consider the wave equation for
a field $A(x,t)$ in the domain $x \in [0,\xi(t)]$ with Dirichlet
boundary conditions, 
{\begin{eqnarray}
\frac{\partial^2 A}{\partial t^2} = c^2 \frac{\partial^2 A}{\partial
x^2},\quad A[0,t] = A[\xi(t),t] = 0,
\end{eqnarray}}\unskip
and the energy associated with this field (up to a dimensional constant),
{\begin{equation}
E_{\mathrm{waves}}(t) = \int_0^{\xi(t)}\left[\frac{1}{2} \left(
\frac{\partial A}{\partial x}\right)^2 + \frac{1}{2c^2} \left(
\frac{\partial A}{\partial t}\right)^2 \right]\,\mathrm{d}x.
\label{eq:e_general}
\end{equation}}\unskip
The time derivative of this energy is computed as
{\begin{eqnarray}
\frac{\mathrm{d} E_{\mathrm{waves}}}{\mathrm{d}t} =
\int_0^{\xi(t)}\left[ \frac{\partial A}{\partial x} \frac{\partial^2
A}{\partial x \partial t} + \frac{1}{c^2}  \frac{\partial A}{\partial
t} \frac{\partial^2 A}{\partial t^2}\right]\,\mathrm{d}x + \dot{\xi}
\left[\frac{1}{2} \left( \frac{\partial A}{\partial
x}\right)^2[\xi(t),t] + \frac{1}{2c^2} \left( \frac{\partial
A}{\partial t}\right)^2[\xi(t),t] \right].
\end{eqnarray}}\unskip
The integral can be computed using integration by part, the wave
equation and the boundary condition at $x=0$, $A[0,t] = 0 \rightarrow
\partial_t A[0,t] = 0$.
{\begin{eqnarray}
\int_0^{\xi(t)}\left[ \frac{\partial A}{\partial x} \frac{\partial^2
A}{\partial x \partial t} + \frac{1}{c^2}  \frac{\partial A}{\partial
t} \frac{\partial^2 A}{\partial t^2}\right]\,\mathrm{d}x&amp;=&amp;
\int_0^{\xi(t)}\left[   \frac{\partial A}{\partial t}  \left(
\frac{1}{c^2} \frac{\partial^2 A}{\partial t^2}   -  \frac{\partial^2
A}{\partial x^2}\right) \right]\,\mathrm{d}x +  \left[   \frac{\partial
A}{\partial t}  \frac{\partial A}{\partial x} \right]_0^{\xi(t)}\Seqnsplit
&amp;=&amp;\left(  \frac{\partial A}{\partial t} \right)[\xi(t),t]  \left( 
\frac{\partial A}{\partial x} \right)[\xi(t),t].
\end{eqnarray}}\unskip
Finally, the second boundary condition results in $A[\xi(t),t] = 0
\rightarrow \partial_t A[\xi(t),t] = - \dot{\xi}\partial_x A[\xi(t),t]
$ and this energy budget reveals the expression of the power generated
by the moving boundary, 
{\begin{eqnarray}
\frac{\mathrm{d} E_{\mathrm{waves}}}{\mathrm{d}t} &amp;=&amp; \left( 
\frac{\partial A}{\partial t} \right)[\xi(t),t]  \left(  \frac{\partial
A}{\partial x} \right)[\xi(t),t] + \dot{\xi} \left[\frac{1}{2} \left(
\frac{\partial A}{\partial x}\right)^2[\xi(t),t] + \frac{1}{2c^2}
\left( \frac{\partial A}{\partial t}\right)^2[\xi(t),t] \right]\Seqnsplit
&amp;=&amp;-\dot{\xi} \frac{(\partial_x A)^2 [\xi(t),t]}{2} \left(1 -
\frac{\dot{\xi}^2}{c^2} \right) \equiv - \dot{\xi} P_{\mathrm{rad}},
\label{eq:prad_general}
\end{eqnarray}}\unskip
hence the expression for the radiation pressure $P_{\mathrm{rad}}$ (up
to the dimensional constant of \eqref{eq:e_general}). If $A$ is the
electromagnetic vector potential, this yields the relativistic
radiation pressure on a moving perfect mirror that will be considered
Section~\ref{S:g_eq_A}, but it also applies to simpler mechanical
systems such that vibrating strings of variable lengths.

\section{Governing equations}\label{S:g_eq}

The considered model is presented in
Figure~\ref{fig:setup}({a}). It consists of waves in a
one-dimensional cavity whose left wall is fixed and whose right wall is
free to respond to the radiation pressure and other external forces.
Even though this setup is generic, it will be described for the
specific framework of electromagnetism in a vacuum, with the walls
assumed to be massive perfect mirrors. The main reasons guiding this
choice are that (i) the wave dynamics in the cavity are fully linear,
(ii) the motion of a mirror does not directly generate waves, and (iii)
the velocity of the massive mechanically moving wall cannot exceed the
wave speed. These stipulations would not hold for acoustic or surface
waves. In this section, different equivalent sets of equations will be
introduced, some useful for concise analytic predictions and others for
simple numerical implementation.

\subsection{In terms of the dimensional electromagnetic fields}

Let $x$ denote the spatial coordinate: the positions of the mirrors are
$0$ (left, fixed wall) and $\xi(t)$ (right, moving wall). A single
linear polarization is considered, corresponding to an electric field
$\mathbf{E}(x,t)= E(x,t) \mathbf{e}_y$ and magnetic field
$\mathbf{B}(x,t)= B(x,t) \mathbf{e}_z$. The propagation and radiation
force of these fields depend on the magnetic permeability of free space
$\mu_0$ and the speed of light $c$. The moving boundary is subject to
the electromagnetic radiation pressure, a linear restoring force, and
potentially other external forces and is therefore characterized, per
unit surface, by its mass $M_s$, its spring constant $K_s$ (of rest
position $L_0$), and additional forces
$\mathbf{F}_{\mathrm{ext}}=F_{\mathrm{ext}}\mathbf{e}_x$. The governing
equations are thus:
{\begin{eqnarray}
&amp; \displaystyle
\frac{\partial E}{\partial x}= - \frac{\partial B}{\partial t}, \quad
\frac{\partial E}{\partial t}= - c^2 \frac{\partial B}{\partial
x},\quad E\left[0,t \right] = 0, \quad E\left[\xi(t),t \right] = \dot{\xi}
B\left[\xi(t),t \right], \label{g_eq:EB1}\Seqnsplit
&amp; \displaystyle
M_s \frac{\mathrm{d}}{\mathrm{d}t} \left( \gamma \dot{\xi} \right) = -
K_s (\xi-L_0) + F_{\mathrm{ext}} + \frac{B\left[\xi(t),t
\right]^2}{2\mu_0} \left( 1- \frac{\dot{\xi}^2}{c^2} \right),
\quad \gamma =\left(1- \frac{\dot{\xi}^2}{c^2}
\right)^{-1/2}.\label{g_eq:EB4}
\end{eqnarray}}\unskip
The boundary conditions \eqref{g_eq:EB1} set the electric field to zero
at the perfect mirrors in their frames of reference. The relativistic
dynamics \eqref{g_eq:EB4} prevent the velocity of the boundary from
reaching $c$. These equations conserve the total energy in the absence
of external forces, 
{\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}t}\left[ M_s \gamma c^2 + K_s
\frac{(\xi-L_0)^2}{2} + \int_{0}^{\xi} \left( \frac{E^2}{2 \mu_0 c^2} +
\frac{B^2}{2 \mu_0} \right)\,\mathrm{d}x\right] = \dot{\xi} 
F_{\mathrm{ext}}.
\end{equation}}\unskip

\subsection{In terms of the dimensionless electromagnetic fields}

The dimensionless fields and variables are defined as
{\begin{eqnarray}
&amp; \displaystyle
x= L_0 \bar{x},\quad t = (L_0/c) \bar{t},\quad \xi(t) = L_0 \bar{\xi}(\bar{t}),\Seqnsplit
&amp; \displaystyle
B(x,t) = \sqrt{\mu_0 K_s L_0}\bar{B}(\bar{x},\bar{t}), \quad E(x,t) =
c\sqrt{\mu_0 K_s L_0}\bar{E}(\bar{x},\bar{t}),\quad F_{\mathrm{ext}}= K_s
L_0 \bar{F}_{\mathrm{ext}}.
\end{eqnarray}}\unskip
They will be considered throughout the remainder of this manuscript;
therefore, I have omitted the over-bars in subsequent equations. The
governing equations and energy budget become
{\begin{eqnarray}
&amp; \displaystyle
\frac{\partial E}{\partial x}= - \frac{\partial B}{\partial t}, \quad
\frac{\partial E}{\partial t}= - \frac{\partial B}{\partial x},
\quad E\left[0,t \right] = 0, \quad E\left[\xi(t),t \right] = \dot{\xi}
B\left[\xi(t),t \right] \label{g_eq:DEB3}\Seqnsplit
&amp; \displaystyle
\alpha^2 \frac{\mathrm{d}}{\mathrm{d}t} \left( \gamma \dot{\xi} \right)
= 1- \xi + F_{\mathrm{ext}} + \frac{B\left[\xi(t),t \right]^2}{2}
\left( 1- \dot{\xi}^2 \right), \quad \gamma =
\left(1-\dot{\xi}^2\right)^{-1/2},\Seqnsplit
&amp; \displaystyle
\frac{\mathrm{d}}{\mathrm{d}t}\left[\alpha^2 \gamma + 
\frac{(\xi-1)^2}{2}+\int_{0}^{\xi} \left( \frac{E^2}{2} + \frac{B^2}{2}
\right)\,\mathrm{d}x\right] = \dot{\xi}  F_{\mathrm{ext}},
\end{eqnarray}}\unskip
where $\alpha = c/ (L_0 \omega)$ and $\omega = \sqrt{K_s/M_s}$ the
natural angular frequency of the boundary.

\subsection{In terms of the dimensionless vector potential}\label{S:g_eq_A}

The theoretical analysis of this system, especially when the motion of
the boundary is prescribed, is facilitated by the introduction of the
dimensionless vector potential $A$ (specifically its $y$-component)
such that $B=\partial_x A$ and $E = - \partial_t A$. The governing
equations are:
{\begin{eqnarray}
\frac{\partial^2 A}{\partial x^2}= \frac{\partial^2 A}{\partial t^2},
\quad A[0,t]=A[\xi(t),t]=0 ,\quad \alpha^2 \frac{\mathrm{d}}{\mathrm{d}t}
\left( \gamma \dot{\xi} \right) = 1- \xi + F_{\mathrm{ext}} +
\frac{(\partial_x A)\left[\xi(t),t \right]^2}{2} \left( 1- \dot{\xi}^2
\right).\label{g_eq:A1}
\end{eqnarray}}\unskip

\subsection{In terms of the dimensionless energy density and flux}\label{S:g_eq_up}

These equations can be expressed in terms of the energy density $u =
(E^2+B^2)/2$ and flux\unskip\break $P=EB$:
{\begin{eqnarray}
&amp; \displaystyle
\frac{\partial u}{\partial t}= - \frac{\partial P}{\partial x}, \quad
\frac{\partial P}{\partial t}= -  \frac{\partial u}{\partial x},
\quad P\left[0,t \right] = 0, \quad P\left[\xi,t \right] = \frac{2
\dot{\xi} u\left[\xi(t),t \right]}{1+ \dot{\xi}^2}, \label{g_eq:up1} \Seqnsplit
&amp; \displaystyle
\alpha^2 \frac{\mathrm{d}}{\mathrm{d}t} \left( \gamma \dot{\xi} \right)
= 1- \xi + F_{\mathrm{ext}} + u\left[\xi(t),t \right] \left( \frac{1-
\dot{\xi}^2}{1+ \dot{\xi}^2} \right) \label{g_eq:up2}.
\end{eqnarray}}\unskip
The electromagnetic energy  no longer involves squared fields and is
given by $E_{\mathrm{em}}=\int_0^\xi u(x,t)\,\mathrm{d}x$.

\section{Numerical model}\label{S:numeric}

Numerical discretization of these equations can be achieved using
finite-difference time-domain (FDTD) on a fixed grid by choosing
adequate boundary conditions and adding and removing grid points as the
boundary moves~\cite{Harfoush_1989, Koutserimpas_2023}. In this study,
I used the more direct approach of introducing stretched coordinates  
on which the governing equations are discretized; thus, the cavity is
always represented by a constant number of grid points. This method
requires no alteration to the boundary conditions and makes energy
conservation straightforward. However, contrary to FDTD, it cannot
generalize to higher spatial dimensions.

\subsection{Governing equations in the stretched domain}

The introduction of the stretched coordinates $(\tilde{t}, \tilde{x}) =
(t, x/\xi(t))$ makes the domain size constant, that is, $x\in[0,\xi(t)]
\rightarrow \tilde{x} \in[0,1]$. Further, the partial derivatives of
any field $f(x,t)=\tilde{f}(\tilde{x},\tilde{t})$ are easily related:
{\begin{eqnarray}
\left( \frac{\partial f}{\partial x}\right)_t =\frac{1}{\xi}  \left(
\frac{\partial \tilde{f}}{\partial
\tilde{x}}\right)_{\tilde{t}},\quad \left( \frac{\partial f}{\partial
t}\right)_x = \left( \frac{\partial \tilde{f}}{\partial
\tilde{t}}\right)_{\tilde{x}}-  \frac{\tilde{x} \dot{\xi} }{\xi} 
\left( \frac{\partial \tilde{f}}{\partial
\tilde{x}}\right)_{\tilde{t}}.
\end{eqnarray}}\unskip
The governing equations reported in Section~\ref{S:g_eq_up} and the
energy budget become
{\begin{eqnarray}
&amp; \displaystyle
\frac{\partial \tilde{u}}{\partial \tilde{t}}=-
\frac{1}{\xi}\frac{\partial \tilde{P}}{\partial \tilde{x}} +
\frac{\tilde{x} \dot{\xi}}{\xi} \frac{\partial \tilde{u}}{\partial
\tilde{x}},\quad\frac{\partial \tilde{P}}{\partial \tilde{t}}=-
\frac{1}{\xi}\frac{\partial \tilde{u}}{\partial \tilde{x}} +
\frac{\tilde{x} \dot{\xi}}{\xi} \frac{\partial \tilde{P}}{\partial
\tilde{x}}, \quad \tilde{P}\left[0,\tilde{t} \right] = 0, \quad
\tilde{P}\left[1,\tilde{t} \right] = \frac{2 \dot{\xi} \tilde{u}
\left[1,\tilde{t} \right]}{1+ \dot{\xi}^2}, \label{g_eq:stretched_up1} \Seqnsplit
&amp; \displaystyle
\alpha^2 \frac{\mathrm{d}}{\mathrm{d}\tilde{t}} \left( \gamma \dot{\xi}
\right) = 1- \xi + F_{\mathrm{ext}} + \tilde{u}\left[1,\tilde{t}
\right] \left( \frac{1- \dot{\xi}^2}{1+ \dot{\xi}^2} \right), 
\quad  \frac{\mathrm{d}}{\mathrm{d}\tilde{t}}\left[\alpha^2 \gamma + 
\frac{(\xi-1)^2}{2}+\xi \int_{0}^{1} \tilde{u}(\tilde{x},\tilde{t})\,
\mathrm{d}\tilde{x}\right] = \dot{\xi}  F_{\mathrm{ext}}.
\label{g_eq:stretched_up2}
\end{eqnarray}}\unskip

\subsection{Spatiotemporal discretization}\label{ST_disc}

These equations are then discretized in space with staggered
finite-difference grids. The discretized energy fluxes $\{
\tilde{P}_n \}_{n \in [0,N]}$ relate to the spatial positions
$\{ \tilde{x}_n= n \Delta X \}_{n \in [0,N]}$ with $\Delta X
= 1/N$, whereas the mesh for the energy densities $\{
\tilde{u}_{n+1/2} \}_{n \in [-1,N]}$ lies at
$\{\tilde{x}_{n+1/2} = (n+1/2) \Delta X\}_{n \in [-1,N]}$.
The derivatives are approximated as
{\begin{eqnarray}
&amp; \displaystyle
\frac{\partial \tilde{u}_{n+1/2}}{\partial \tilde{t}}=-
\frac{1}{\xi}\left(\frac{\tilde{P}_{n+1}-\tilde{P}_{n}}{\Delta
X}\right) + \left( \frac{\tilde{x}_{n+1/2} \dot{\xi}}{\xi}\right)\left(
\frac{\tilde{u}_{n+3/2}-\tilde{u}_{n-1/2}}{2 \Delta X}\right), \quad n
\in [0,N-1]\label{num_e1}\Seqnsplit
&amp; \displaystyle
\frac{\partial \tilde{P}_n}{\partial \tilde{t}}=- \frac{1}{\xi}
\left(\frac{\tilde{u}_{n+1/2}-\tilde{u}_{n-1/2}}{\Delta X}\right)
+\left( \frac{\tilde{x}_n \dot{\xi}}{\xi}\right)\left(
\frac{\tilde{P}_{n+1}-\tilde{P}_{n-1}}{2 \Delta
X}\right),\quad n \in [1,N-1]\label{num_e2}\Seqnsplit
&amp; \displaystyle
\tilde{P}_0 = 0, \quad \tilde{P}_N = \frac{\dot{\xi}
(\tilde{u}_{N-1/2}+\tilde{u}_{N+1/2})}{1+ \dot{\xi}^2},\quad \alpha^2
\frac{\mathrm{d}}{\mathrm{d}\tilde{t}} \left( \gamma \dot{\xi} \right)
= 1- \xi + F_{\mathrm{ext}} +
\frac{(\tilde{u}_{N-1/2}+\tilde{u}_{N+1/2})}{2} \left( \frac{1-
\dot{\xi}^2}{1+ \dot{\xi}^2} \right).\label{num_e4}\quad
\end{eqnarray}}\unskip
Prescribing the ghost nodes $\tilde{u}_{-1/2}=\tilde{u}_{1/2}$ and
$\tilde{u}_{N+1/2}=\tilde{u}_{N-1/2}$ enforces the discrete energy
budget, 
{\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}\tilde{t}}\left[\alpha^2 \gamma + 
\frac{(\xi-1)^2}{2}+\xi \sum_{n\in[0,N-1]} \tilde{u}_{n+1/2} \Delta
X\right] = \dot{\xi}  F_{\mathrm{ext}}.
\end{equation}}\unskip
Notably, although this spatial discretization conserves the total
energy, the propagation becomes dispersive and the energy density is no
longer guarantied to remain positive (these features become more
apparent with energy variations at small scales $O(\Delta X)$).
Finally, Equations~\eqref{num_e1}--\eqref{num_e4} are numerically
implemented and advanced in time using a fourth-order Runge--Kutta
scheme. All the simulations reported in this article were performed
with $N=10^5$ grid points, unless stated otherwise, and a fixed
timestep $\Delta t = 1/(2N)$.

\section{Limit cases}\label{S:simple_limits}

\subsection{Waves in a harmonically modulated cavity}

Here, a few simple cases will be illustrated using the numerical model.
The first considers the canonical problem of waves in a cavity of
harmonically modulated length. It is  mainly presented as an
introduction to the phenomenology of waves in a modulated domain but
also serves as a test case for our numerical method and as an
introduction to the theoretical recurrence\unskip\break relations.

\subsubsection{Recurrence relations}

Recurrence relations can be derived from the vector potential equations
obtained in Section~\ref{S:g_eq_A}. They characterize the back-and-forth
propagation of waves in the cavity, along with their change in
amplitude as they reflect off the moving boundary. The general solution
of the wave equation \eqref{g_eq:A1} is
{\begin{equation}
A(x,t) = A_+(x+t) + A_-(x-t).\label{rec_rel_interm_0}
\end{equation}}\unskip
The boundary conditions are enforced with these new functions, then
differentiated with\unskip\break time:
{\begin{eqnarray}
&amp; \displaystyle
A \left[\xi(t),t \right] = 0 =A_+ \left[\xi(t)+t\right] + A_-
\left[\xi(t)-t\right] \quad\Longrightarrow \dot{A}_+ \left[\xi(t)+t\right]
= \left(\frac{1- \dot{\xi}(t)}{1+ \dot{\xi}(t)} \right) \dot{A}_-
\left[\xi(t)-t\right],\label{rec_rel_interm_1}\Seqnsplit
&amp; \displaystyle
A (0,t) = 0 = A_+(t) + A_- (-t) \Longrightarrow \dot{A}_+(t) =
\dot{A}_-(-t) \quad\Longrightarrow \partial_xA(0,t) = 2 \dot{A}_+(t) =
2 \dot{A}_-(-t).\label{rec_rel_interm_2}
\end{eqnarray}}\unskip
Since $t$ is arbitrary in \eqref{rec_rel_interm_2}, this results in both
{\begin{eqnarray}
\partial_xA \left[0,t+\xi(t) \right] = 2 \dot{A}_+ \left[t+\xi(t)
\right], \quad\partial_xA \left[0,t-\xi(t) \right] = 2 \dot{A}_-
\left[\xi(t)-t \right].\label{rec_rel_interm_3}
\end{eqnarray}}\unskip
Combining \eqref{rec_rel_interm_1} and \eqref{rec_rel_interm_3} gives 
{\begin{equation}
\partial_x A\left[0, t + \xi(t) \right] = \left(\frac{1-
\dot{\xi}(t)}{1+ \dot{\xi}(t)} \right)\partial_x A\left[0, t - \xi(t)
\right].\label{rec_rel_interm_4}
\end{equation}}\unskip
Equation \eqref{rec_rel_interm_4} provides a relation between the
fields at the left (stationnary) boundary before ($t-\xi(t)$) and after
($t + \xi(t)$) they reflect off the moving mirror (at time $t$). This
equation is well-known and can be found in several instances in the
literature (Equations (7) and (8) in Ref.~\cite{Meplan_1996}). To
express this result in terms of the variables used in our numerical
formulation, I introduce the radiation pressure $\mathcal{P}$ acting on
the moving boundary and the energy density $u_L$ measured on the left
boundary  from \eqref{g_eq:A1}:
{\begin{eqnarray}
\mathcal{P}(t) = \frac{\partial_x A\left[\xi(t),t
\right]^2}{2}(1-\xi(t)^2),\quad u_L(t \pm \xi(t)) = \frac{\partial_x A
\left[0,t \pm \xi(t)\right]^2}{2}.
\end{eqnarray}}\unskip
Differentiating \eqref{rec_rel_interm_0} with respect to $x$ and then
\eqref{rec_rel_interm_3} and \eqref{rec_rel_interm_4}, I obtain
{\begin{eqnarray}
\partial_x A\left[\xi(t),t \right]= \dot{A}_+\left[\xi(t)+t \right]  +
\dot{A}_-\left[\xi(t)-t \right] = \frac{ \partial_xA \left[0,t+\xi(t)
\right] + \partial_xA \left[0,t-\xi(t) \right]}{2} = \frac{\partial_x
A\left[0,t-\xi(t) \right]}{1+ \dot{\xi}(t)}.\nonumber\\ 
\end{eqnarray}}\unskip
Finally, recurrence relations on both $u_L$ and $\mathcal{P}$ are 
{\begin{eqnarray}
\mathcal{P}(t) = \left(\frac{1- \dot{\xi}(t)}{1+ \dot{\xi}(t)}
\right)u_L\left[t-  \xi(t) \right], \quad u_L\left[t+  \xi(t) \right] =
\left(\frac{1- \dot{\xi}(t)}{1+ \dot{\xi}(t)} \right)^2u_L\left[t- 
\xi(t) \right]. \label{rec_rel_final_UP}
\end{eqnarray}}\unskip
These hold irrespective of the specific motion of the boundary
$\xi(t)$. They track the propagation of the energy density at the left
wall at time $t-\xi(t)$ such that they relate (i) to the radiation
pressure generated at time $t$ when the wave meets the mirror with
position $\xi(t)$, and (ii) to the new energy density modified by
Doppler effect that returns to the left wall at time $t+\xi(t)$.  

\subsubsection{Divergence of the energy at $\omega \simeq c\rmpi/L_0$}

Assume that an external force drives the left boundary according to
$\xi(t) =L_0\left[1+ a \sin(\omega t) \right]$, i.e., that with
dimensionless variables $\xi(t) = 1+ a \sin(\alpha^{-1} t)$ ($0&lt;a &lt;1$
and $a  &lt;\alpha$). Thus, if $\alpha^{-1} = \rmpi$, that is, if $\omega =
c \rmpi / L_0$ is the first non-zero eigenfrequency of the
electromagnetic cavity at rest, the recurrence relation
\eqref{rec_rel_final_UP} for $u_L$ at times $t_n=2n$ (such that
$\xi(t_n)=1$ and $\xi'(t_n) = a \rmpi$) becomes
{\begin{equation}
u_L(2n+1) =\left(\frac{1+ a \rmpi}{1- a \rmpi} \right)^2 u_L(
2n-1)=\left(\frac{1+ a \rmpi}{1- a \rmpi} \right)^4 u_L( 2n-3)= \cdots =
\left(\frac{1+ a \rmpi}{1- a \rmpi} \right)^{2n} u_L(1),
\end{equation}}\unskip
representing energy divergence. The same procedure can be adapted for
an $\alpha^{-1}$ that slightly differs from $\rmpi$. This divergence
persists in the range~\cite{Dittrich_1994}
{\begin{equation}
\frac{\omega L_0}{c} \in \left[ \frac{\rmpi}{1+a},  \frac{\rmpi}{1-a}
\right].\label{Dittrich_instability_range}
\end{equation}}\unskip
Numerical simulations were performed with the same initial conditions
and boundary motion as in Ref.~\cite{Dittrich_1994}: $u(x,0)=P(x,0) =
\exp [-60 (x-0.5)^2]$ and $\xi(t) = 1+ a \sin (\alpha^{-1} t)$ with
$a=0.2$. I refer the reader to the additional movies
\textsf{Movie\_inva2} ($\alpha^{-1}=2$) and \textsf{Movie\_inva3}
($\alpha^{-1}=3$) (Supplementary information). The second movie clearly
depicts cumulative Doppler effects that dramatically increase the
energy of a pulse propagating back and forth. The energy of the waves
as a function of $\alpha^{-1}$ and $t$ is shown in
Figure~\ref{fig:dittrich}(left); it reproduces the instability range of
\eqref{Dittrich_instability_range} and coincides with the theoretical
Figure 3 of Ref.~\cite{Dittrich_1994}.

\begin{figure}
\includegraphics{fig02} 
\caption{\label{fig:dittrich}(Left) Energy $E_{\mathrm{em}}$ in a cavity of
modulated length $\xi(t) = 1+a \sin(\alpha^{-1} t)$. The initial
conditions $u(x,0)=P(x,0) = \exp [-60 (x-0.5)^2]$ and specific value of
$a=0.2$ are consistent with Ref.~\cite{Dittrich_1994}. The
dimensionless angular frequency $\alpha^{-1} = \omega L_0/c$ varies in
increments of 0.01. The vertical dashed lines correspond to the
theoretical instability range \eqref{Dittrich_instability_range}.
(Right) Energy density at the left wall $\epsilon^2
\phi(n,t)=u_L(t+2n)$ in a cavity whose right wall responds to the
radiation pressure ($n$ is the number of interactions between the pulse
and the moving boundary). The initial conditions are $u(x,t) = -P(x,t)
= \alpha^2  \exp [-60 (x-0.5)^2 ]$, $\xi(0)=1$ and $\dot{\xi}(0)=0$,
with $\alpha = 5\times 10^{-3}$. The dashed lines are the theoretical
predictions of \eqref{burgers}.} 
{\vspace*{7pt}} 
\end{figure}

\subsection{Waves in a cavity a with critically damped oscillating boundary} 

Here, the motion of the boundary is no longer externally prescribed but
driven by radiation pressure. A two-way coupling develops between the
waves and the moving mirror, which renders theoretical predictions
based on recurrence relations \eqref{rec_rel_final_UP} much more
challenging because $\xi$ now depends on the entire history of
$\mathcal{P}$. Thereafter, I focus on a specific limit in which this
memory effect is restricted and can be explicitly accounted for: in
addition to radiation pressure and linear restoring forces, the
boundary is considered subject to critical damping, i.e.,
$F_{\mathrm{ext}} = - 2\alpha \dot{\xi}$.  This dissipation makes the
mirror relax on a timescale $\alpha$ that is assumed to be small
compared with unity, that is, much faster than the time required for a
pulse to travel back and forth in the cavity. 

\subsubsection{Slow modulation of small-amplitude pulses}\label{subs:burgers}

The dynamics of the boundary are
{\begin{equation}
\alpha^2 \frac{\mathrm{d}}{\mathrm{d}t} \left( \gamma \dot{\xi} \right)
= 1- \xi - 2 \alpha \dot{\xi} +
\mathcal{P}(t).\label{model_1_pfd_relat}
\end{equation}}\unskip
Assuming small-amplitude fields and a fast oscillator timescale
$\alpha$, which results in small displacements of the moving mirror, I
expand 
{\begin{eqnarray}
\mathcal{P}(t) = \epsilon^2 p(t),\quad \xi(t) = 1+ \epsilon^2 \eta(t),\quad 
u_L(t) = \epsilon^2 \varphi(t),\label{overdamped_scaling}
\end{eqnarray}}\unskip
with $\alpha = \epsilon \ll 1$. Equation \eqref{model_1_pfd_relat}, at
leading orders, models a fast critically damped oscillator, 
{\begin{equation}
\ddot{\eta} + \frac{2 \dot{ \eta}}{\epsilon} + \frac{\eta}{\epsilon^2}
= p(t) + O(\epsilon^2),
\end{equation}}\unskip
the displacement of which is computed with a Green's function, 
{\begin{equation}
\eta(t) = \int_0^\infty s e^{-s/\epsilon}P(t-s)\,\mathrm{d}s = 
\int_0^\infty s e^{-s/\epsilon} \left[P(t) - s\dot{P}(t) + \cdots
\right]\,\mathrm{d}s = P(t) - 2 \epsilon \dot{P}(t) + O(\epsilon^2).
\end{equation}}\unskip
The recurrence relation \eqref{rec_rel_final_UP} for $\mathcal{P}$ 
yields $p(t) = \varphi(t-1) + O(\varepsilon^2)$ and the position of the
mirror directly relates to the electromagnetic energy at the other end
of the cavity some time before,
{\begin{equation}
\eta(t) = \varphi(t-1) - 2 \epsilon \dot{\varphi}(t-1) +
O(\varepsilon^2).\label{model_1_eta}
\end{equation}}\unskip
Equation \eqref{rec_rel_final_UP} for $u_L$  that connects two
successive reflections of the pulse on the left wall becomes
{\begin{equation}
\varphi\left[t + 1 + \epsilon^2 \eta(t) \right] = \left[1 - 4
\epsilon^2 \dot{\eta}(t) + O(\epsilon^4) \right] \varphi\left[t-1 -
\varepsilon^2 \eta(t) \right].\label{model_1_phi}
\end{equation}}\unskip
The successive bounces of a single pulse can be disentangled in this
limit with $\phi(n,t) = \varphi(t+2n)$ with $t \in[0,2]$ and any
integer $n$. This leads, with \eqref{model_1_eta} and
\eqref{model_1_phi}, to 
{\begin{equation}
\partial_n \phi + 6 \epsilon^2 \phi  \partial_t \phi = 4 \epsilon^3
\left[ (\partial_t \phi)^2 + 2 \phi \partial_{tt} \phi \right] +
O(\epsilon^4),\label{burgers}
\end{equation}}\unskip
which is a regularized Burgers equation. At order $\epsilon^2$, the
total electromagnetic energy is conserved and it reduces to the strict
Burgers equation, known to generate shock from any localized
pulse~\cite{Whitham2011}. Regularization and damping arise at the next
order.

Consistent with \eqref{overdamped_scaling}, the simulations were
initialized with $\xi(0)=1$, $\dot{\xi}(0)=0$ and $u(x,0) = -P(x,0) =
\alpha^2  \exp [-60 (x-0.5)^2 ]$. The movie
\textsf{Movie3\_alpha0p2} (Supplementary information) illustrates the
general dynamics and definition of the quantity of interest $\epsilon^2
\phi(n,t)=u_L(t+2n)$ for $\alpha = 0.2$. However, this value of
$\alpha$ is too large for \eqref{burgers} to hold.  The results shown
in  Figure~\ref{fig:dittrich}(right) for $\alpha = 5\times 10^{-3}$ 
are in good agreement with the analysis derived in the limit of $\alpha
\rightarrow 0$, except close to the peak of the pulse where
comparatively fast dynamics occur that require higher order
corrections.

\subsection{Shocks of very short pulses}\label{sec:shocks}

I now consider an initial condition with a pulse of very short duration
compared to the response time of the critically damped oscillator. A
similar formal analysis can be performed, that reveals that the
radiation pressure is mainly balanced by inertia during these brief
interactions. Conservation laws directly capture the electromagnetic
energy loss during this  shock, which is transferred to the mirror and
eventually dissipated before the next interaction takes place. 

Consider an electromagnetic pulse of dimensionless energy $E_i$ and
momentum $P_i=E_i$ traveling towards $x \rightarrow \infty$, that
reflects off a comparatively slow oscillator initially at rest. Its
final energy $E_f$ corresponds to a momentum $P_f = -E_f$ because it
then progresses towards $x \rightarrow - \infty$.  The velocity of the
mirror just after the pulse leaves is denoted as $\dot{\xi}_f$ (the
position of the mirror is assumed to be unchanged). The conservation of
energy and momentum are therefore
{\begin{eqnarray}
E_i = E_f + \alpha^2 (\gamma_f -1),\quad P_i = P_f + \alpha^2 \gamma_f
\dot{\xi}_f\Longrightarrow E_i = - E_f + \alpha^2 \gamma_f \dot{\xi}_f,
\end{eqnarray}}\unskip
where $\gamma_f = (1- \dot{\xi}_f^2)^{-1/2}$. This system can be solved
for $\dot{\xi}_f$ and yields 
{\begin{eqnarray}
\frac{E_f}{E_i} =  \frac{1}{1+\mu}, \quad \gamma_f -1 =
\frac{\mu^2}{2+\mu}, \quad \mu =  \frac{2 E_i}{\alpha^2}.\label{eq_shock}
\end{eqnarray}}\unskip
The initial condition $u(x,0) = P(x,0) =10  \exp [- (x-0.5)^2 /
\sigma_x^2 ]$ and $\xi(0)= \dot{\xi}(0)= 0$ with $\sigma_x=10^{-3}$ and
$\alpha=0.2$ illustrates this regime. Here, I refer the reader to the
movie \textsf{Movie4\_sx0p001} (Supplementary information) that shows
the dynamics up to $t=5$. The various components of the energy are
shown in Figure~\ref{Fig:shock}(left). The electromagnetic energy
$E_{\mathrm{em}}=\int_0^\xi u(x,t)\,\mathrm{d}x$ is almost piece-wise constant
(with values $\{ E_{\mathrm{em},n} \}_{n}$), some of it being
transferred as kinetic  $E_{\mathrm{k}}(t) =  \alpha^2 \gamma(t)$ then potential
$E_{\mathrm{p}}(t) = (\xi(t)-1)^2/2$ energies at each shock. The squared
restitution coefficient $E_f/E_i$ of \eqref{eq_shock} quantitatively
compares to the DNS (see Figure~\ref{Fig:shock}(right)).

\begin{figure}
{\vspace*{-1pt}}
\includegraphics{fig03}
{\vspace*{-1pt}}
\caption{\label{Fig:shock}(Left) Energy in a cavity in which a pulse
$u(x,0)= P(x,0)=10  \exp [-10^{6} (x-0.5)^2]$  interacts with a wall of
a comparatively slow characteristic time. Rapid variations in the
electromagnetic energy are observed, that can be modeled as shocks.
(Right) Ratio of the final to initial electromagnetic energy for the
first 25 shocks compared to \eqref{eq_shock}.}
{\vspace*{-1pt}}
\end{figure}

\section{Numerical simulations in the conservative limit}\label{S:DNS}

\subsection{Evolution of a single pulse}\label{sec:recurrences}

The fully conservative regime, in which the moving boundary is only
subject to the radiation pressure and a linear restoring force
($F_{\mathrm{ext}}=0$), represents a considerable theoretical
challenge. In this section, I report numerical simulations that
evidence the general dynamics, first considering the evolution of a
single pulse of dimensionless energy 0.02 in a cavity initially at
rest, i.e., 
{\begin{equation}
u(x,0)=P(x,0)=0.02\sqrt{\frac{60}{\rmpi}} \exp \left[-60(x-0.5)^2\right],
\quad \xi(0)=1, \quad\dot{\xi}(0)=0.
\end{equation}}\unskip
Energy conservation implies that the displacement of the boundary is
bounded, $\vert \xi(t) -1 \vert \leqslant 0.2$. The evolution of the
electromagnetic energy for $\alpha^{-1} \in  [2, 4.5 ]$ experiences
periodic oscillations, as shown in 
Figure~\ref{fig:single_pulse_free_cavity}(left). For deeper insights, a
longer run was computed for $\alpha^{-1}=3.27$ of up to 100 time units:
see movie \textsf{Movie5\_inva3p27} (Supplementary information) and 
Figure~\ref{fig:single_pulse_free_cavity}(right) that show the
evolution of the initial and almost final electromagnetic energy
$u(x,0)$ and $u(x,99.5)$.  Although the pulse shapes develop sharp
variations, the periodic evolution of the electromagnetic energy
persists (it accounts for 99.99\% of the total energy at $t=99.5$).
These slow recurrences are reminiscent of the
Fermi--Pasta--Ulam--Tsingou (FPUT) problem in which the nonlinearity
comes from the wave equation rather than coupling with a moving
boundary, resulting in an extremely long time for thermal equilibrium
to be reached~\cite{FPU_55}.

\begin{figure}
\includegraphics{fig04} 
\caption{\label{fig:single_pulse_free_cavity}(Left) Energy $E_{\mathrm{em}}$ in
a cavity whose left wall can oscillate with a natural angular frequency
$\omega$. The initial conditions $u(x,0)=P(x,0) =0.02 \sqrt{60/\rmpi}
\exp [-60 (x-0.5)^2]$ are similar to those of
Figure~\ref{fig:dittrich}(left).  The dimensionless angular frequency
$\alpha^{-1} = \omega L_0/c$ varies in increments of 0.01 and the
vertical dashed line corresponds to $\alpha^{-1}=3.27$. (Right) Energy
density in the cavity at times $t=0$ and $t=99.5$
($\alpha^{-1}=3.27$).} 
\end{figure}

\subsection{Equilibrium spectra} \label{sec:eq_spectra}

As a more direct route to steady-state convergence, the numerical
simulation is initialized with random electromagnetic noise in a cavity
in which the moving boundary is at rest at time $t=0$, 
{\begin{equation}
\tilde{u}_{n+1/2}(0)=E_{\mathrm{em}}(0) 
\frac{\tilde{a}_{n+1/2}}{\sum_{m=0}^{N-1}\tilde{a}_{m+1/2}\Delta X},
\quad P(x,0)=0, \quad \xi(0)=1, \quad\dot{\xi}(0)=0,
\end{equation}}\unskip
where $\{ \tilde{a}_{n+1/2}\}$ are random values drawn from
the uniform distribution on the open interval $(0,1)$ and $E_{\mathrm{em}}(0)$
is the total energy in the system. The evolution of the spectral
electromagnetic components in the cavity are difficult to compute using
a spatial Fourier transform because the domain size evolves. A more
convenient approach characterizes the temporal power spectrum density
(PSD) of the energy density variations at the left (fixed) wall,
$u(0,t)- \langle u(0,t) \rangle_t$ ($\langle \cdot \rangle_t$ denotes a
time average over the range in which the PSD is computed). Simulations
were run with $N=100$ grid points, $\alpha^{-1}=\rmpi$, and three
different values of $E_{\mathrm{em}}(0)= \{ 0.002, 0.01, 0.05 \}$, up
to $4 \times 10^8$ time units.

The supplemental movie \textsf{Movie6\_random\_IC\_N100\_E0p05} depicts
the dynamics at both the initial and final times for $E_{\mathrm{em}}(0) =
0.05$: the electromagnetic field clearly evolves toward a state that no
longer displays high temporal or spatial frequencies. This is also
evident from time series of $u(0,t)$ shown in 
Figure~\ref{fig:random_noise}(left). This feature is observed in all
three simulations. The PSDs at the final time are, for small values of
the total energy $E_{\mathrm{em}}(0)$, almost restricted to the first
eigenfrequencies of the linear system ($\{  f_n = n/2 \}_{n
\geqslant 1}$, see Figure~\ref{fig:random_noise}(right)). As the total
energy increases, the peak corresponding to $f_1 = 1/2$ splits in $f_1
\pm f_c/2$ because of nonlinear coupling with the moving boundary: the
very low frequency $f_c$, also observed in the PSDs, is found to slowly
modulate the oscillation amplitude of $\xi(t)$ and to scale with the
total energy as $f_c = 3.8 E_{\mathrm{em}}$. This relation has been confirmed
for a larger number of grid points $N$ ($N=200$ and $E_{\mathrm{em}}= 0.05$).
Even though the extinction of the high frequencies may be caused by
numerical dispersion, these results nevertheless support the
possibility of nonlinear steady states with very simple spectral
structures in which the waves are strongly coupled with the motion of
the moving boundary.

\begin{figure}
\includegraphics{fig05} 
\caption{\label{fig:random_noise}(Left) Energy density at the left
(fixed) wall in a cavity initialized with random noise of total energy
$E_{\mathrm{em}}(0) = 0.05$ with $N=100$ grid points. (Right) PSDs of the energy
density fluctuations at the left wall at final time $t \in [ 4 \times
10^8, 500+4 \times 10^8 ]$ for $E_{\mathrm{em}}(0)= \{ 0.002, 0.05
\}$.}
\end{figure}

\section{Conclusion}

The two-way coupling between linear waves trapped in a cavity and a
fast moving boundary undergoing large displacements was introduced in
the context of classical electromagnetism. No practical experimental
setup can follow this model because it involves perfect mirrors moving
at a fraction of the speed of light (the only plausible relevance is in
astrophysics, at the surface of highly conductive neutron
stars~\cite{Neutron_star}). However, it provides a convenient framework
to evidence emerging features such as the generation of sharp profiles
Section~\ref{subs:burgers}, finite restitution coefficients for fast
waves Section~\ref{sec:shocks}, slow recurrences
Section~\ref{sec:recurrences}, nonlinear steady-states in which energy
is concentrated in a few modes Section~\ref{sec:eq_spectra}. These
observations can guide the understanding of experiments with mechanical
waves, such as water waves generated by a paddle in a container or
acoustic waves forced by a piezoelectric device in a cavity. It shows
that qualitative behavior usually ascribed to the bulk non-linearities
can also result from a \textit{linear} propagation coupled with a
moving boundary.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that  could benefit from this article, and have
declared no affiliations other than their research organizations.

\back{}

\section*{Supplementary data}

Supporting information for this article is available on the journal's
website under \printDOI\ or from the author. 
%Supplementary material for this article is supplied as a separate archive movies.zip.

\CDRsupplementaryTwotypes[application/zip]{supplementary-material}{\cdrattach{crphys-242-suppl.zip}}

\printbibliography
\refinput{crphys20250150-reference.tex}

\end{document}
</pre></body></html>