1–99

# Green’s functions, propagation invariants, reciprocity theorems, wave-field
representations and propagator matrices in 2D time-dependent materials

Kees Wapenaar
Department of Geoscience and Engineering Delft University of Technology  The
Netherlands

(2024)

###### keywords:

Time-dependent material, Green’s function, Representation, Propagator matrix.

††volume: 999

The study of wave propagation and scattering in time-dependent materials is a
rapidly growing field of research. Whereas for 1D applications there is a
simple relation between the wave equations for space-dependent and time-
dependent materials, this relation is less straightforward for multi-
dimensional materials.

This paper discusses fundamental aspects of 2D electromagnetic and acoustic
wave propagation and scattering in homogeneous, time-dependent materials. This
encompasses a review of transmission and reflection at a single time boundary,
a discussion of the Green’s function and its symmetry properties in a
piecewise continuous time-dependent material, a discussion of propagation
invariants (including the net field-momentum density), general reciprocity
theorems, and wave field representations.

Analogous to the well-known expression for Green’s function retrieval by time-
correlation of passive measurements in a space-dependent material, an
expression is derived for Green’s function retrieval by space-correlation of
passive measurements in a time-dependent material.

The paper concludes with the discussion of the propagator matrix for a
piecewise continuous time-dependent material, its symmetry properties and its
relation with the Green’s function.

##  1 Introduction

Temporal changes of material parameters have an effect on wave propagation and
scattering, comparable with, but not identical to, the effect of spatial
changes. Since the initial work on wave propagation and scattering in time-
dependent materials (Morgenthaler, 1958; Jiang, 1975), the research in this
field has recently got significant momentum (Bacot et al., 2016; Fink & Fort,
2017; Koutserimpas & Fleury, 2018; Bal et al., 2019; Peng et al., 2020; Caloz
& Deck-Léger, 2020a; Mounaix et al., 2020; Ramaccia et al., 2020; Garnier,
2021; Apffel & Fort, 2022; Moussa et al., 2023; Hidalgo-Caballero et al.,
2023; Ptitcyn et al., 2023). Various authors discuss the analogy between the
wave equations for time-dependent and space-dependent materials (Mendonça &
Shukla, 2002; Xiao et al., 2014; de Hoop & Lager, 2014; Salem & Caloz, 2015;
Torrent et al., 2018; Caloz & Deck-Léger, 2020b; Van Manen et al., 2024). For
1D applications, the roles of the time- and space-coordinates are interchanged
between the wave equations for both types of material. However, causality
conditions apply to the time coordinate in both types of material and hence
they are not interchanged. As a consequence, solutions of the wave equation in
a time-dependent material are, in general, not one-to-one exchangeable with
those in a space-dependent material. In a recent paper (Wapenaar et al., 2024)
we systematically analyze the similarities and differences of waves in 1D
space-dependent and time-dependent materials. In multi-dimensional materials,
the number of space dimensions (two or three) is different from the number of
time dimensions (one) and hence the analogy between the wave equations for
both types of material is less straightforward. In the current paper we
discuss fundamental aspects of wave propagation and scattering in 2D time-
dependent materials. The discussion partly reviews work discussed in the
references mentioned above but also contains new results. Like in our paper on
1D materials, we use a unified notation which simultaneously captures
electromagnetic, acoustic and elastodynamic shear waves. We discuss
transmission and reflection coefficients, Green’s functions, propagation
invariants, reciprocity theorems, wave field representations, Green’s function
retrieval, propagator matrices and the relation between the Green’s function
and the propagator matrix of a time-dependent material.

##  2 Unified wave equation

We consider 2D wave propagation in the $x,z$-plane, assuming the material
parameters, the sources and the wave fields are independent of the
$y$-coordinate. We denote position in this plane by the Cartesian coordinate
vector ${\bf x}=(x,z)$ and time by $t$. We capture electromagnetic waves
(transverse electric and transverse magnetic), acoustic waves and horizontally
polarized shear waves by the following unified equations (Carcione &
Cavallini, 1995; de Hoop, 1995; Wapenaar et al., 2001; Carcione & Robinson,
2002; Burns et al., 2020)

$\displaystyle\partial_{t}U+{\bf\nabla}\cdot{\bf Q}$ $\displaystyle=$
$\displaystyle a,$ (1) $\displaystyle\partial_{t}{\bf V}+{\bf\nabla}P$
$\displaystyle=$ $\displaystyle{\bf b},$ (2)

where $\partial_{t}$ stands for the partial differential operator
$\frac{\partial}{\partial t}$ and ${\bf\nabla}$ is the 2D nabla operator,
defined as ${\bf\nabla}=(\partial_{x},\partial_{z})$. Furthermore, $U({\bf
x},t)$, ${\bf V}({\bf x},t)$, $P({\bf x},t)$ and ${\bf Q}({\bf x},t)$ are
space- and time-dependent wave-field quantities and $a({\bf x},t)$ and ${\bf
b}({\bf x},t)$ are space- and time-dependent source quantities. The boldface
quantities ${\bf V}$, ${\bf Q}$ and ${\bf b}$ denote vectors with two
components, hence ${\bf V}=(V_{x},V_{z})$, etc. All quantities are further
specified in Table 1 for the different wave phenomena considered in this
paper. For electromagnetic waves, equations (1) and (2) are Maxwell’s
equations in the $x,z$-plane (Landau & Lifshitz, 1960; de Hoop, 1995). For
example, for transverse electric waves (row 1 in Table 1) we have ${\bf
Q}=(H_{z},-H_{x})$, hence, ${\bf\nabla}\cdot{\bf
Q}=\partial_{x}H_{z}-\partial_{z}H_{x}=-({\bf\nabla}\times{\bf H})_{y}$, which
is minus the $y$-component of the curl of the magnetic field vector ${\bf
H}({\bf x},t)$. Hence, equation (1) reads for this situation
$\partial_{t}D_{y}-({\bf\nabla}\times{\bf H})_{y}=-J_{y}^{\rm e}$. With some
reordering, equation (2) can be shown to be the 2D version of
$\partial_{t}{\bf B}+{\bf\nabla}\times{\bf E}=-{\bf J}^{m}$. In a similar way
it can be shown that for transverse magnetic waves (row 2 in Table 1),
equations (1) and (2) are again Maxwell’s equations in the $x,z$-plane, but in
reversed order. For acoustic waves (row 3 in Table 1) we have
${\bf\nabla}\cdot{\bf Q}=\partial_{x}v_{x}+\partial_{z}v_{z}$, which is the
divergence of the particle velocity vector ${\bf v}({\bf x},t)$. Hence, for
this situation equation (1) stands for
$-\partial_{t}\Theta+{\bf\nabla}\cdot{\bf v}=q$, which is the acoustic
deformation equation, and equation (2) becomes $\partial_{t}{\bf
m}+{\bf\nabla}p={\bf f}$, which quantifies equilibrium of momentum (de Hoop,
1995). Similarly, for horizontally polarized shear waves (row 4 in Table 1),
equation (1) stands for equilibrium of momentum and equation (2) is the 2D
elastic deformation equation.

Table 1:  Specification of the quantities in equations (1) – (4). For TE (transverse electric) and TM (transverse magnetic) waves, the quantities are electric and magnetic flux densities ${D}$ and ${B}$, electric and magnetic field strengths ${E}$ and ${H}$, permittivity $\varepsilon$, permeability $\mu$, and external electric and magnetic current densities ${J}^{\rm e}$ and ${J}^{\rm m}$. For acoustic waves they are cubic dilatation $\Theta$, mechanical momentum density ${m}$, acoustic pressure $p$, particle velocity ${v}$, compressibility $\kappa$, mass density $\rho$, volume-injection rate density $q$ and external force density ${f}$. For SH (horizontally polarised shear) waves, $m$, $v$, $\rho$ and $f$ are defined the same as for acoustic waves, and the additional quantities are strain $e$, stress $\tau$, shear modulus $\mu$ and external deformation rate density $h$. | $U$ | $V_{x}$ | $V_{z}$ | $P$ | $Q_{x}$ | $Q_{z}$ | $\alpha$ | $\beta$ | $a$ | $b_{x}$ | $b_{z}$
---|---|---|---|---|---|---|---|---|---|---|---
1\. TE | $D_{y}$ | $B_{z}$ | $-B_{x}$ | $E_{y}$ | $H_{z}$ | $-H_{x}$ | $\varepsilon$ | $\mu$ | $-J_{y}^{\rm e}$ | $-J_{z}^{\rm m}$ | $J_{x}^{\rm m}$
2\. TM | $B_{y}$ | $-D_{z}$ | $D_{x}$ | $H_{y}$ | $-E_{z}$ | $E_{x}$ | $\mu$ | $\varepsilon$ | $-J_{y}^{\rm m}$ | $J_{z}^{\rm e}$ | $-J_{x}^{\rm e}$
3\. Acoustic | $-\Theta$ | $m_{x}$ | $m_{z}$ | $p$ | $v_{x}$ | $v_{z}$ | $\kappa$ | $\rho$ | $q$ | $f_{x}$ | $f_{z}$
4\. SH | $m_{y}$ | $-2e_{yx}$ | $-2e_{yz}$ | $v_{y}$ | $-\tau_{yx}$ | $-\tau_{yz}$ | $\rho$ | $\frac{1}{\mu}$ | $f_{y}$ | $2h_{yx}$ | $2h_{yz}$

The wave-field quantities are mutually related via the constitutive equations,
as follows

$\displaystyle U$ $\displaystyle=$ $\displaystyle\alpha P,$ (3)
$\displaystyle{\bf V}$ $\displaystyle=$ $\displaystyle\beta{\bf Q},$ (4)

where $\alpha$ and $\beta$ are material parameters. They are also specified in
Table 1 for the different wave phenomena. We can use these equations to
eliminate two of the four wave field quantities from equations (1) and (2).

Let us first consider the common situation of a material which is space-
dependent and constant in time, with parameters $\alpha({\bf x})$ and
$\beta({\bf x})$. This implies (using equations (3) and (4))
$\partial_{t}U=\partial_{t}(\alpha P)=\alpha\partial_{t}P$ and
$\partial_{t}{\bf V}=\partial_{t}(\beta{\bf Q})=\beta\partial_{t}{\bf Q}$.
Hence, for this situation elimination of $U$ and ${\bf V}$ from equations (1)
and (2) yields

$\displaystyle\alpha\partial_{t}P+{\bf\nabla}\cdot{\bf Q}$ $\displaystyle=$
$\displaystyle a,$ (5) $\displaystyle\beta\partial_{t}{\bf Q}+{\bf\nabla}P$
$\displaystyle=$ $\displaystyle{\bf b}.$ (6)

This well-known system of equations for the wave field quantities $P$ and
${\bf Q}$ underlies wave propagation in inhomogeneous, time-independent
materials.

In this paper we consider homogeneous, time-dependent materials, with
parameters $\alpha(t)$ and $\beta(t)$. For this situation we cannot use
equations (5) and (6). Instead, using equations (3) and (4), we now have
${\bf\nabla}P={\bf\nabla}(\frac{1}{\alpha}U)=\frac{1}{\alpha}{\bf\nabla}U$ and
${\bf\nabla}\cdot{\bf Q}={\bf\nabla}\cdot(\frac{1}{\beta}{\bf
V})=\frac{1}{\beta}{\bf\nabla}\cdot{\bf V}$. Hence, elimination of $P$ and
${\bf Q}$ from equations (1) and (2) yields

$\displaystyle\partial_{t}U+\frac{1}{\beta}{\bf\nabla}\cdot{\bf V}$
$\displaystyle=$ $\displaystyle a,$ (7) $\displaystyle\partial_{t}{\bf
V}+\frac{1}{\alpha}{\bf\nabla}U$ $\displaystyle=$ $\displaystyle{\bf b}.$ (8)

This system of equations for the wave field quantities $U$ and ${\bf V}$
underlies wave propagation in homogeneous, time-dependent materials. They form
the basis for the discussion in the remainder of this paper. For wave
propagation in materials that are simultaneously space- and time-dependent,
see Innanen (2018); Caloz & Deck-Léger (2020b); Apffel & Fort (2022); Van
Manen et al. (2024); Ammari et al. (2024). For non-reciprocal wave propagation
in modulated metamaterials, we refer to Lurie (2007); Willis (2011, 2012);
Trainiti & Ruzzene (2016); Haberman & Guild (2016); Nassar et al. (2017b, a);
Goldsberry et al. (2019); Sotoodehfar et al. (2023).

Equations (7) and (8) are used to derive various fundamental properties of
wave propagation and scattering in time-dependent materials. It is also useful
to consider a second order wave equation for the scalar field $U({\bf x},t)$.
This equation is obtained by eliminating ${\bf V}({\bf x},t)$ from equations
(7) and (8), which yields

$\displaystyle\partial_{t}(\beta\partial_{t}U)-\beta c^{2}{\bf\nabla}^{2}U=s,$
(9)

with $c(t)$ being the time-dependent propagation velocity, defined as

$\displaystyle c=\frac{1}{\sqrt{\alpha\beta}},$ (10)

$s({\bf x},t)$ being the source function, defined as

$\displaystyle s=\partial_{t}(\beta a)-{\bf\nabla}\cdot{\bf b},$ (11)

and
${\bf\nabla}^{2}={\bf\nabla}\cdot{\bf\nabla}=\partial_{x}^{2}+\partial_{z}^{2}$.
Finally, we define a time-dependent quantity $\eta(t)$ as

$\displaystyle\eta=\sqrt{\frac{\beta}{\alpha}}=\beta c=\frac{1}{\alpha c}.$
(12)

For TE and acoustic waves (rows 1 and 3 in Table 1), $\eta$ stands for
impedance, whereas it stands for admittance for TM and SH waves (rows 2 and 4
in Table 1).

##  3 Transmission and reflection of plane waves at a time boundary

###  3.1 Transmission and reflection coefficients

We review transmission and reflection coefficients for a monochromatic plane
wave that is incident on a time boundary (Xiao et al., 2014; Morgenthaler,
1958; Mendonça & Shukla, 2002; Caloz & Deck-Léger, 2020b). For convenience we
define the time boundary at $t=0$. For $t<0$ we assume constant parameters
$\alpha_{1}$ and $\beta_{1}$. Similarly, for $t>0$ we assume constant
parameters $\alpha_{2}$ and $\beta_{2}$. For $t<0$, the incident monochromatic
plane wave is described in complex notation by

$\displaystyle U_{I}({\bf x},t)$ $\displaystyle=$ $\displaystyle\exp\\{i({\bf
k}_{1}\cdot{\bf x}-\omega_{1}t)\\},$ (13)

where subscript $I$ stands for incident, $i$ is the imaginary unit,
$\omega_{1}$ is the angular frequency of the incident wave and ${\bf k}_{1}$
is its wave vector, defined as ${\bf k}_{1}=(k_{x,1},k_{z,1})$. For
convenience we consider a unit amplitude and zero phase at ${\bf x}={\bf 0}$
and $t=0$ (where ${\bf 0}=(0,0)$). The propagation angle (relative to the
$z$-axis) is equal to $\operatorname{atan2}(k_{x,1},k_{z,1})$. According to
equation (9) for the source-free situation (i.e., for $s=0$), $U_{I}({\bf
x},t)$ should obey $\partial_{t}^{2}U_{I}=c_{1}^{2}{\bf\nabla}^{2}U_{I}$, with
$c_{1}=1/\sqrt{\alpha_{1}\beta_{1}}$. This implies
$\omega_{1}^{2}=c_{1}^{2}|{\bf k}_{1}|^{2}$ or, choosing positive square-roots
on both sides, $\omega_{1}=c_{1}|{\bf k}_{1}|$.

The boundary conditions state that $U$ and ${\bf V}$ are continuous over the
time boundary (Xiao et al., 2014; Morgenthaler, 1958; Mendonça & Shukla, 2002;
Caloz & Deck-Léger, 2020b; de Hoop & Lager, 2014), hence, we also need an
expression for ${\bf V}_{I}$. From equations (8) (with ${\bf b}={\bf 0})$ and
(13) we obtain

$\displaystyle{\bf V}_{I}({\bf x},t)$ $\displaystyle=$
$\displaystyle\eta_{1}\frac{{\bf k}_{1}}{|{\bf k}_{1}|}\exp\\{i({\bf
k}_{1}\cdot{\bf x}-\omega_{1}t)\\},$ (14)

with $\eta_{1}=1/(\alpha_{1}c_{1})$. Unlike in the situation of scattering at
a space boundary, where the reflected wave propagates in the same half-space
as the incident wave, in the situation of a time boundary there is no
reflected wave before the time boundary, since this would violate causality.
The reflected and transmitted waves both exist only after the time boundary.
For $t>0$, the transmitted monochromatic plane wave is described by

$\displaystyle U_{T}({\bf x},t)$ $\displaystyle=$ $\displaystyle
T_{u}\exp\\{i({\bf k}_{2}\cdot{\bf x}-\omega_{2}t)\\},$ (15)
$\displaystyle{\bf V}_{T}({\bf x},t)$ $\displaystyle=$ $\displaystyle
T_{u}\eta_{2}\frac{{\bf k}_{2}}{|{\bf k}_{2}|}\exp\\{i({\bf k}_{2}\cdot{\bf
x}-\omega_{2}t)\\},$ (16)

with $T_{u}$ denoting the transmission coefficient for the wave-field quantity
$U$. Furthermore, $\omega_{2}$ and ${\bf k}_{2}=(k_{x,2},k_{z,2})$ are the
angular frequency and wave vector of the transmitted wave. They are related
via $\omega_{2}=c_{2}|{\bf k}_{2}|$, with
$c_{2}=1/\sqrt{\alpha_{2}\beta_{2}}$. Finally, $\eta_{2}=1/(\alpha_{2}c_{2})$.
The reflected wave for $t>0$ is described by

$\displaystyle U_{R}({\bf x},t)$ $\displaystyle=$ $\displaystyle
R_{u}\exp\\{i({\bf k}_{2}\cdot{\bf x}+\omega_{2}t)\\},$ (17)
$\displaystyle{\bf V}_{R}({\bf x},t)$ $\displaystyle=$ $\displaystyle-
R_{u}\eta_{2}\frac{{\bf k}_{2}}{|{\bf k}_{2}|}\exp\\{i({\bf k}_{2}\cdot{\bf
x}+\omega_{2}t)\\},$ (18)

with $R_{u}$ denoting the reflection coefficient for the wave-field quantity
$U$. Note the opposite signs in front of the terms $\omega_{2}t$ for the
transmitted and reflected waves, which implies these waves propagate in
opposite directions. Since $U({\bf x},t)$ and ${\bf V}({\bf x},t)$ are
continuous at $t=0$, we have

$\displaystyle U_{I}({\bf x},0)$ $\displaystyle=$ $\displaystyle U_{T}({\bf
x},0)+U_{R}({\bf x},0),$ (19) $\displaystyle{\bf V}_{I}({\bf x},0)$
$\displaystyle=$ $\displaystyle{\bf V}_{T}({\bf x},0)+{\bf V}_{R}({\bf x},0).$
(20)

These equations should hold for all ${\bf x}$, which implies ${\bf k}_{1}={\bf
k}_{2}$. From this it follows that the angular frequency of the transmitted
and reflected waves is different from that of the incident wave, according to
$\omega_{2}=\frac{c_{2}}{c_{1}}\omega_{1}$ (Apffel & Fort, 2022; Mendonça &
Shukla, 2002; Caloz & Deck-Léger, 2020b). Substituting the appropriate
expressions into equations (19) and (20) yields

$\displaystyle 1$ $\displaystyle=$ $\displaystyle T_{u}+R_{u},$ (21)
$\displaystyle\eta_{1}$ $\displaystyle=$ $\displaystyle\eta_{2}(T_{u}-R_{u}).$
(22)

From this we obtain the following expressions for the transmission and
reflection coefficients

$\displaystyle T_{u}$ $\displaystyle=$
$\displaystyle\frac{\eta_{2}+\eta_{1}}{2\eta_{2}},$ (23) $\displaystyle R_{u}$
$\displaystyle=$ $\displaystyle\frac{\eta_{2}-\eta_{1}}{2\eta_{2}}.$ (24)

Note that, unlike for the situation of a space boundary, these coefficients
are independent of the wave vector and hence of the propagation angle.

By multiplying the left- and right-hand sides of equation (21) with those of
equation (22), we obtain the following relation between the transmission and
reflection coefficients

$\displaystyle\eta_{1}=\eta_{2}(T_{u}^{2}-R_{u}^{2}).$ (25)

Finally, we define transmission and reflection coefficients $T_{p}$ and
$R_{p}$ for the wave-field quantity $P$. Using equation (3), we obtain
$T_{p}=\frac{\alpha_{1}}{\alpha_{2}}T_{u}$ and
$R_{p}=\frac{\alpha_{1}}{\alpha_{2}}R_{u}$, or

$\displaystyle T_{p}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\biggl{(}\frac{\alpha_{1}}{\alpha_{2}}+\frac{c_{2}}{c_{1}}\biggr{)},$
(26) $\displaystyle R_{p}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\biggl{(}\frac{\alpha_{1}}{\alpha_{2}}-\frac{c_{2}}{c_{1}}\biggr{)}.$
(27)

These expressions are consistent with those given by Xiao et al. (2014).

###  3.2 Conservation of net field-momentum density

For electromagnetic waves, we define the net power-flux density as $j({\bf
x},t)=\frac{1}{2}\Re({\bf E}^{*}\times{\bf H})\cdot{\bf n}$, with ${\bf
n}={\bf k}/|{\bf k}|$ being the normal vector in the direction of the wave
vector ${\bf k}$, the asterisk $*$ denoting complex conjugation and $\Re$
denoting the real part. For transverse electric waves (row 1 in Table 1) this
reduces to $j({\bf
x},t)=\frac{1}{2}\Re(E_{y}^{*}H_{z}n_{x}-E_{y}^{*}H_{x}n_{z})$, and for
transverse magnetic waves (row 2 in Table 1) to $j({\bf
x},t)=\frac{1}{2}\Re(-H_{y}^{*}E_{z}n_{x}+H_{y}^{*}E_{x}n_{z})$. Both cases
are captured by the compact definition

$\displaystyle j({\bf x},t)=\frac{1}{2}\Re(P^{*}{\bf Q})\cdot{\bf n}.$ (28)

This equation also captures the net power-flux density for acoustic and
horizontally polarized shear waves (rows 3 and 4 in Table 1).

Next to the net power-flux density, we define the net field-momentum density
for electromagnetic waves as $M({\bf x},t)=\Re({\bf D}^{*}\times{\bf
B})\cdot{\bf n}$ (Burns et al., 2020; Feynman et al., 1963). For transverse
electric and transverse magnetic waves this can be written as

$\displaystyle M({\bf x},t)=\Re(U^{*}{\bf V})\cdot{\bf n}.$ (29)

This equation also captures the net field-momentum density for acoustic and
horizontally polarized shear waves (which should not be confused with the
mechanical momentum density $m$ mentioned in Table 1).

We analyse the quantities $j({\bf x},t)$ and $M({\bf x},t)$ for monochromatic
plane waves before and after a time boundary (equations (13) – (18)).
Substituting $U=U_{I}$, ${\bf V}={\bf V}_{I}$ for $t<0$ and $U=U_{T}+U_{R}$,
${\bf V}={\bf V}_{T}+{\bf V}_{R}$ for $t>0$ into equation (29) we find, using
equation (25), that $M({\bf x},t)$ is constant for all ${\bf x}$ and $t$.
Hence, the net field-momentum density is conserved when passing a time
boundary. Using equations (3), (4) and (10), it follows that $j({\bf x},t)$
and $M({\bf x},t)$ are mutually related via $j({\bf
x},t)=\frac{c^{2}(t)}{2}M({\bf x},t)$. Hence,
$j_{2}=\frac{c_{2}^{2}}{c_{1}^{2}}j_{1}$, where $j_{1}$ and $j_{2}$ are the
net power-flux densities before and after the time boundary. Hence, the net
power-flux density is not conserved when passing a time boundary. This is the
result of energy being added to or extracted from the wave field by the
mechanism that modulates the material parameters (Morgenthaler, 1958; Mendonça
& Shukla, 2002; Caloz & Deck-Léger, 2020b).

##  4 Green’s function

We consider again a homogeneous, time-dependent material, with parameters
$\alpha(t)$ and $\beta(t)$. We introduce the Green’s function ${\cal G}({\bf
x},{\bf x}_{0},t,t_{0})$ as the response to an impulsive point source at ${\bf
x}={\bf x}_{0}$ at time instant $t=t_{0}$. Hence, it obeys wave equation (9),
with the source term on the right-hand side replaced by an impulsive point
source, according to

$\displaystyle\partial_{t}(\beta\partial_{t}{\cal G})-\beta
c^{2}{\bf\nabla}^{2}{\cal G}=\delta({\bf x}-{\bf x}_{0})\delta(t-t_{0}),$ (30)

where $\delta(t)$ is the Dirac delta function. Furthermore, the Green’s
function obeys the causality condition

$\displaystyle{\cal G}({\bf x},{\bf x}_{0},t,t_{0})=0\quad\mbox{for}\quad
t<t_{0}.$ (31)

Assuming that the material is time-independent beyond an arbitrary large but
finite time, this causality condition implies that ${\cal G}({\bf x},{\bf
x}_{0},t,t_{0})$ is outward propagating for $|{\bf x}|\to\infty$. Since the
material is homogeneous, the Green’s function is shift-invariant, hence

$\displaystyle{\cal G}({\bf x},{\bf x}_{0},t,t_{0})={\cal G}({\bf x}-{\bf
x}_{0},{\bf 0},t,t_{0}).$ (32)

Figure 1: Snapshots of the Green’s function ${\cal G}({\bf x},{\bf 0},t,0)$
(convolved with a spatial wavelet) for a piecewise constant time-dependent
material, with a time boundary at $t=50.0\,\mu$s. Movie available at
https://www.keeswapenaar.nl/TimeMaterial/Green.mp4.

Figure 2: Cross-sections at $x=0$ of the Green’s function of Figure 1, i.e.,
${\cal G}({\bf x},{\bf 0},t,0)|_{x=0}$.

We discuss a numerical example of ${\cal G}({\bf x},{\bf x}_{0},t,t_{0})$ for
a piecewise constant material with a single time boundary, with propagation
velocities $c_{1}=1500$ m/s for $t<50\,\mu$s and $c_{2}=2500$ m/s for
$t>50\,\mu$s (the parameter $\beta$ is constant). We choose the source at
${\bf x}_{0}={\bf 0}$ and $t_{0}=0$. Figure 1 shows snapshots of ${\cal
G}({\bf x},{\bf 0},t,0)$ (convolved with a 2D spatial wavelet with a central
wavenumber $k_{0}/2\pi=200$ m-1). The first three frames show an expanding
circular wave front, up to the time boundary at $t=50\,\mu$s. In the following
frames the wave field is split into a transmitted wave front, which expands
further, and a reflected wave front, which propagates back and collapses to a
focus at ${\bf x}={\bf 0}$ and $t=79.75\,\mu$s. At exactly $t=80\,\mu$s the
focused field is zero (not shown) and changes its sign, after which it
continues as an expanding circular wave front with opposite amplitude, as
shown in the last frame. Hence, the focus at ${\bf x}={\bf 0}$ and
$t=80\,\mu$s acts as a virtual source for a wave field in a material with
propagation velocity $c_{2}=2500$ m/s. Figure 2 shows cross-sections at $x=0$
of these snapshots in the form of a $z,t$-diagram (i.e., for ${\bf x}=(0,z)$,
as if receivers were placed along a vertical line in the middle of these
snapshots, through the source position). Left of the green line at $t_{0}=0$
the field is zero, conform the causality condition of equation (31). Right of
this green line, direct waves propagate away from the source, in two
directions (up and down), until they reach the time boundary at $t=50\,\mu$s,
indicated by the dashed blue line. Right of this blue line, transmitted and
reflected waves can be seen. The reflected waves propagate back and focus at
the position of the original source at $z=0$.

Whereas the Green’s function ${\cal G}({\bf x},{\bf x}_{0},t,t_{0})$ is the
response to an impulsive point source $\delta({\bf x}-{\bf
x}_{0})\delta(t-t_{0})$ for arbitrary ${\bf x}_{0}$ and $t_{0}$ (equation
(30)), the wave field $U({\bf x},t)$ is the response to a source distribution
$s({\bf x},t)$ (equation (9)). We obtain a simple representation for the wave
field $U$ in terms of ${\cal G}$, assuming they both reside in the same medium
and both are outward propagating for $|{\bf x}|\to\infty$. Since both
equations are linear, we obtain the representation for $U({\bf x},t)$ by
applying Huygens’ superposition principle, according to (Morse & Feshbach,
1953; Bleistein, 1984)

$\displaystyle U({\bf x},t)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}{\rm d}t_{0}\int_{{\mathbb{R}}^{2}}{\cal
G}({\bf x},{\bf x}_{0},t,t_{0})s({\bf x}_{0},t_{0}){\rm d}{\bf x}_{0},$ (33)

where ${\bf x}_{0}$ and $t_{0}$ are variables and where ${\mathbb{R}}$ is the
set of real numbers. This is a special case of the more general representation
derived in section 8. Using equation (32), we can rewrite equation (33) as

$\displaystyle U({\bf x},t)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}{\rm d}t_{0}\int_{{\mathbb{R}}^{2}}{\cal
G}({\bf x}-{\bf x}_{0},{\bf 0},t,t_{0})s({\bf x}_{0},t_{0}){\rm d}{\bf
x}_{0}.$ (34)

Note that the space integral represents a 2D space convolution. Since the
material is time-dependent, the time integral cannot be rewritten as a genuine
time convolution.

Figure 3: The wave field $U({\bf x},t)$ in response to a distributed source in
the same piecewise constant time-dependent material as in Figure 1. Movie
available at https://www.keeswapenaar.nl/TimeMaterial/TUD.mp4.

Figure 3 is an illustration of equation (34) for a source distribution $s({\bf
x},t)=s_{0}({\bf x})\delta(t)$, i.e., for a distributed source $s_{0}$ as a
function of ${\bf x}$ at $t=0$. The first frame shows the response at
$t=0.25\,\mu$s to this source in the same time-dependent material as in the
previous example. The next frames show the evolution of this response over
time. After the time boundary at $t=50\,\mu$s, the transmitted field continues
expanding, whereas the reflected field propagates back and collapses to a
reproduction of the distributed source at $t=79.75\,\mu$s. After this, the
focused field changes its sign (at $t=80\,\mu$s) and expands again. Results
like this have been obtained with physical experiments by Bacot et al. (2016);
Fink & Fort (2017); Peng et al. (2020).

Finally, for the special case of a time-independent medium, the analytical
solution of equation (30), with causality condition (31), reads

$\displaystyle{\cal G}({\bf x},{\bf x}_{0},t,t_{0})=\frac{1}{2\pi\beta
c^{2}}\frac{H\Bigl{(}t-t_{0}-\frac{|{\bf x}-{\bf
x}_{0}|}{c}\Bigr{)}}{\sqrt{(t-t_{0})^{2}-\frac{|{\bf x}-{\bf
x}_{0}|^{2}}{c^{2}}}},$ (35)

where $H(t)$ is the Heaviside step function. The direct waves in Figure 2 can
be seen as an illustration of this expression for ${\cal G}({\bf x},{\bf
0},t,0)|_{x=0}$, convolved with a spatial wavelet.

##  5 Unified wave equation and Green’s function in wave-vector time domain

###  5.1 Unified wave equation in wave-vector time domain

We define the following 2D Fourier transformation

$\displaystyle\check{U}({\bf k},t)=\int_{{\mathbb{R}}^{2}}U({\bf
x},t)\exp(-i{\bf k}\cdot{\bf x}){\rm d}{\bf x},$ (36)

with wave vector ${\bf k}$ defined as ${\bf k}=(k_{x},k_{z})$. For the inverse
2D Fourier transformation we obtain

$\displaystyle U({\bf
x},t)=\frac{1}{4\pi^{2}}\int_{{\mathbb{R}}^{2}}\check{U}({\bf k},t)\exp(i{\bf
k}\cdot{\bf x}){\rm d}{\bf k}.$ (37)

Equation (37) expresses $U({\bf x},t)$ as a superposition of “space-harmonic”
plane waves $\exp(i{\bf k}\cdot{\bf x})$ with complex amplitudes
$\check{U}({\bf k},t)$. Hence, equation (36) can be interpreted as a
decomposition of the field $U({\bf x},t)$ into space-harmonic plane waves.

Applying the nabla operator ${\bf\nabla}$ to both sides of equation (37)
yields

$\displaystyle{\bf\nabla}U({\bf
x},t)=\frac{1}{4\pi^{2}}\int_{{\mathbb{R}}^{2}}i{\bf k}\check{U}({\bf
k},t)\exp(i{\bf k}\cdot{\bf x}){\rm d}{\bf k}.$ (38)

Hence, the operation ${\bf\nabla}$ in the space-time domain corresponds to
multiplication by $i{\bf k}$ in the wave-vector time domain. Using this
property, we obtain for the Fourier transforms of equations (7) and (8)

$\displaystyle\partial_{t}\check{U}+\frac{1}{\beta}i{\bf k}\cdot\check{\bf V}$
$\displaystyle=$ $\displaystyle\check{a},$ (39)
$\displaystyle\partial_{t}\check{\bf V}+\frac{1}{\alpha}i{\bf k}\check{U}$
$\displaystyle=$ $\displaystyle\check{\bf b},$ (40)

with time-dependent material parameters $\alpha(t)$ and $\beta(t)$. The second
order wave equation (9) transforms to

$\displaystyle\partial_{t}(\beta\partial_{t}\check{U})+\beta c^{2}|{\bf
k}|^{2}\check{U}=\check{s},$ (41)

with $|{\bf k}|^{2}={\bf k}\cdot{\bf k}=k_{x}^{2}+k_{z}^{2}$, and the source
function $\check{s}({\bf k},t)$ defined as

$\displaystyle\check{s}=\partial_{t}(\beta\check{a})-i{\bf k}\cdot\check{\bf
b}.$ (42)

###  5.2 Green’s function in wave-vector time domain

The transformed Green’s function $\check{\cal G}({\bf k},{\bf x}_{0},t,t_{0})$
obeys

$\displaystyle\partial_{t}(\beta\partial_{t}\check{\cal G})+\beta c^{2}|{\bf
k}|^{2}\check{\cal G}=\exp(-i{\bf k}\cdot{\bf x}_{0})\delta(t-t_{0}),$ (43)

with causality condition

$\displaystyle\check{\cal G}({\bf k},{\bf
x}_{0},t,t_{0})=0\quad\mbox{for}\quad t<t_{0}.$ (44)

For the special case of a time-independent medium we have

$\displaystyle\check{\cal G}({\bf k},{\bf x}_{0},t,t_{0})=\exp(-i{\bf
k}\cdot{\bf x}_{0})H(t-t_{0})\frac{\sin(|{\bf k}|c(t-t_{0}))}{\eta|{\bf k}|}.$
(45)

In Appendix A it is shown that the inverse Fourier transform of equation (45)
is the space-time domain Green’s function of equation (35).

##  6 Propagation invariants for piecewise continuous materials

###  6.1 Conservation of net field-momentum density

Analogous to equations (28) and (29), we define the net power-flux density and
the net field-momentum density in the ${\bf k},t$-domain as

$\displaystyle\check{\jmath}({\bf k},t)=\frac{1}{2}\Re(\check{P}^{*}\check{\bf
Q})\cdot{\bf n}$ (46)

and

$\displaystyle\check{M}({\bf k},t)=\Re(\check{U}^{*}\check{\bf V})\cdot{\bf
n},$ (47)

respectively, with ${\bf n}={\bf k}/|{\bf k}|$ (but note that
$\check{\jmath}({\bf k},t)$ and $\check{M}({\bf k},t)$ are not the spatial
Fourier transforms of $j({\bf x},t)$ and $M({\bf x},t)$).

Propagation invariants in a time-dependent material are wave-field-related
quantities that remain constant over time. We start by showing that for a
source-free, time-dependent material, the net field-momentum density defined
in equation (47) is a propagation invariant. To this end we analyze
$\partial_{t}\check{M}({\bf
k},t)=\frac{1}{2}\partial_{t}(\check{U}^{*}\check{\bf V}+\check{\bf
V}^{*}\check{U})\cdot{\bf n}$. Applying the product rule for differentiation,
using equations (39) and (40) for the source-free situation and the definition
${\bf n}={\bf k}/|{\bf k}|$, we obtain

$\displaystyle\partial_{t}\check{M}({\bf k},t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigl{(}(\partial_{t}\check{U})^{*}\check{\bf
V}+\check{U}^{*}\partial_{t}\check{\bf V}+(\partial_{t}\check{\bf
V})^{*}\check{U}+\check{\bf V}^{*}\partial_{t}\check{U}\Bigr{)}\cdot{\bf
k}/|{\bf k}|$ $\displaystyle=$
$\displaystyle\frac{1}{2}\biggl{(}\Bigl{(}\frac{1}{\beta}i{\bf
k}\cdot\check{\bf V}^{*}\Bigr{)}\check{\bf
V}-\check{U}^{*}\Bigl{(}\frac{1}{\alpha}i{\bf
k}\check{U}\Bigr{)}+\Bigl{(}\frac{1}{\alpha}i{\bf
k}\check{U}^{*}\Bigr{)}\check{U}-\check{\bf V}^{*}\Bigl{(}\frac{1}{\beta}i{\bf
k}\cdot\check{\bf V}\Bigr{)}\biggr{)}\cdot{\bf k}/|{\bf k}|=0.$

This expression implies that $\check{M}({\bf k},t)$ is a propagation
invariant, or in other words, the net field-momentum density is a conserved
quantity. The derivation above holds for continuously varying material
parameters. However, $\check{U}$ and $\check{\bf V}\cdot{\bf n}$ are
continuous over a time boundary and so is $\check{M}({\bf k},t)$. Combining
these results, it follows that $\check{M}({\bf k},t)$ is a propagation
invariant for a time-dependent material with piecewise continuous parameters.

From equations (3), (4), (10), (46) and (47) we obtain

$\displaystyle\check{\jmath}({\bf k},t)=\frac{c^{2}(t)}{2}\check{M}({\bf
k},t),$ (49)

which implies that for a time-dependent medium with piecewise continuous
parameters the net power-flux density is not a propagation invariant, similar
as noted in section 3.2 for $j({\bf x},t)$ over a single time boundary.

###  6.2 General propagation invariants

Next, we derive more general propagation invariants. We consider two mutually
independent states (i.e., solutions of equations (39) and (40)), which we
distinguish with subscripts $A$ and $B$. We analyze the quantities
$\partial_{t}(\check{U}_{A}\check{\bf V}_{B}-\check{\bf
V}_{A}\check{U}_{B})\cdot(i{\bf k})$ and
$\partial_{t}(\check{U}_{A}^{*}\check{\bf V}_{B}+\check{\bf
V}_{A}^{*}\check{U}_{B})\cdot(i{\bf k})$. The latter can be seen as a
generalization of $2\partial_{t}\check{M}({\bf
k},t)=\partial_{t}(\check{U}^{*}\check{\bf V}+\check{\bf
V}^{*}\check{U})\cdot{\bf n}$, with subscripts $A$ and $B$ added and ${\bf
n}={\bf k}/|{\bf k}|$ replaced by $i{\bf k}$ (the latter replacement
facilitates the inverse Fourier transformations in later sections). Applying
the product rule for differentiation, using equations (39) and (40) for states
$A$ and $B$, yields

$\displaystyle\partial_{t}(\check{U}_{A}\check{\bf V}_{B}-\check{\bf
V}_{A}\check{U}_{B})\cdot(i{\bf k})$ $\displaystyle=$ $\displaystyle|{\bf
k}|^{2}(\alpha_{B}^{-1}-\alpha_{A}^{-1})\check{U}_{A}\check{U}_{B}+(\beta_{B}^{-1}-\beta_{A}^{-1})\check{V}_{{\rm
d},A}\check{V}_{{\rm d},B}$ (50) $\displaystyle+\check{a}_{A}\check{V}_{{\rm
d},B}-\check{b}_{{\rm d},A}\check{U}_{B}-\check{V}_{{\rm
d},A}\check{a}_{B}+\check{U}_{A}\check{b}_{{\rm d},B},$

and

$\displaystyle\partial_{t}(\check{U}_{A}^{*}\check{\bf V}_{B}+\check{\bf
V}_{A}^{*}\check{U}_{B})\cdot(i{\bf k})$ $\displaystyle=$ $\displaystyle|{\bf
k}|^{2}(\alpha_{B}^{-1}-\alpha_{A}^{-1})\check{U}_{A}^{*}\check{U}_{B}+(\beta_{B}^{-1}-\beta_{A}^{-1})\check{V}_{{\rm
d},A}^{*}\check{V}_{{\rm d},B}$ (51)
$\displaystyle+\check{a}_{A}^{*}\check{V}_{{\rm d},B}-\check{b}_{{\rm
d},A}^{*}\check{U}_{B}-\check{V}_{{\rm
d},A}^{*}\check{a}_{B}+\check{U}_{A}^{*}\check{b}_{{\rm d},B},$

respectively, with

$\displaystyle\check{V}_{\rm d}$ $\displaystyle=$ $\displaystyle i{\bf
k}\cdot\check{\bf V},$ (52) $\displaystyle\check{b}_{\rm d}$ $\displaystyle=$
$\displaystyle i{\bf k}\cdot\check{\bf b},$ (53)

for states $A$ and $B$. The subscript ${\rm d}$ in $\check{V}_{\rm d}$ and
$\check{b}_{\rm d}$ refers to “divergence” (since $i{\bf k}\cdot\check{\bf V}$
and $i{\bf k}\cdot\check{\bf b}$ are the Fourier transforms of
${\nabla}\cdot{\bf V}$ and ${\nabla}\cdot{\bf b}$). Following similar
arguments as below equation (6.1), equations (50) and (51) hold for a time-
dependent material with piecewise continuous parameters.

Note that in equations (50) and (51) not only the wave fields are labeled with
subscripts $A$ and $B$, but also the source terms and the material parameters
(which may be different in the two states). In the next section we continue
with the general expressions of equations (50) and (51). Here we consider the
special case that sources are absent and the material parameters are the same
in both states, which implies that the right-hand sides of equations (50) and
(51) vanish. Hence, for this situation it follows that
$(\check{U}_{A}\check{\bf V}_{B}-\check{\bf V}_{A}\check{U}_{B})\cdot(i{\bf
k})$ and $(\check{U}_{A}^{*}\check{\bf V}_{B}+\check{\bf
V}_{A}^{*}\check{U}_{B})\cdot(i{\bf k})$ are propagation invariants for a
homogeneous time-dependent material. They are the counterparts of propagation
invariants for an inhomogeneous time-independent material (Haines, 1988;
Kennett et al., 1990; Koketsu et al., 1991; Takenaka et al., 1993; Wapenaar et
al., 2004), which are useful for the analysis of symmetry properties of
transmission and reflection responses and for the design of efficient
numerical modelling schemes.

##  7 Reciprocity theorems

###  7.1 General reciprocity theorems

Reciprocity theorems interrelate wave fields in two different states
(Rayleigh, 1878; Lorentz, 1895; Knopoff & Gangi, 1959; de Hoop, 1966). Here we
use equations (50) and (51) to derive reciprocity theorems for a homogeneous
time-dependent material with piecewise continuous parameters $\alpha(t)$ and
$\beta(t)$. Integrating equations (50) and (51) with respect to time, taking
into account that $\check{U}$ and $\check{V}_{\rm d}=i{\bf k}\cdot\check{\bf
V}$ are continuous at time instants where $\alpha(t)$ and $\beta(t)$ are
discontinuous, we obtain

$\displaystyle(\check{U}_{A}\check{V}_{{\rm d},B}-\check{V}_{{\rm
d},A}\check{U}_{B})\Bigr{|}_{t_{b}}^{t_{e}}$ $\displaystyle=$
$\displaystyle\int_{t_{b}}^{t_{e}}\Bigl{(}|{\bf
k}|^{2}(\alpha_{B}^{-1}-\alpha_{A}^{-1})\check{U}_{A}\check{U}_{B}+(\beta_{B}^{-1}-\beta_{A}^{-1})\check{V}_{{\rm
d},A}\check{V}_{{\rm d},B}$ (54) $\displaystyle+\check{a}_{A}\check{V}_{{\rm
d},B}-\check{b}_{{\rm d},A}\check{U}_{B}-\check{V}_{{\rm
d},A}\check{a}_{B}+\check{U}_{A}\check{b}_{{\rm d},B}\Bigr{)}{\rm d}t$

and

$\displaystyle(\check{U}_{A}^{*}\check{V}_{{\rm d},B}-\check{V}_{{\rm
d},A}^{*}\check{U}_{B})\Bigr{|}_{t_{b}}^{t_{e}}$ $\displaystyle=$
$\displaystyle\int_{t_{b}}^{t_{e}}\Bigl{(}|{\bf
k}|^{2}(\alpha_{B}^{-1}-\alpha_{A}^{-1})\check{U}_{A}^{*}\check{U}_{B}+(\beta_{B}^{-1}-\beta_{A}^{-1})\check{V}_{{\rm
d},A}^{*}\check{V}_{{\rm d},B}$ (55)
$\displaystyle+\check{a}_{A}^{*}\check{V}_{{\rm d},B}-\check{b}_{{\rm
d},A}^{*}\check{U}_{B}-\check{V}_{{\rm
d},A}^{*}\check{a}_{B}+\check{U}_{A}^{*}\check{b}_{{\rm d},B}\Bigr{)}{\rm
d}t,$

respectively, with subscripts $b$ and $e$ standing for “begin” and “end”.
Products like $\check{U}_{A}\check{U}_{B}$ in the ${\bf k},t$-domain
correspond to spatial convolutions in the ${\bf x},t$-domain. Therefore we
call equation (54) the reciprocity theorem of the space-convolution type. It
is the counterpart of the reciprocity theorem of the time-convolution type for
an inhomogeneous time-independent medium (de Hoop, 1995; Fokkema & van den
Berg, 1993). On the other hand, products like $\check{U}_{A}^{*}\check{U}_{B}$
in the ${\bf k},t$-domain correspond to spatial correlations in the ${\bf
x},t$-domain. Therefore we call equation (55) the reciprocity theorem of the
space-correlation type. It is the counterpart of the reciprocity theorem of
the time-correlation type for an inhomogeneous time-independent medium (de
Hoop, 1995; Bojarski, 1983).

We use equation (54) in section 8 to derive a general wave field
representation and in section 9 we use equation (55) to derive an expression
for Green’s function retrieval, both for time-dependent materials. Here we
discuss two special cases.

###  7.2 Source-receiver reciprocity

First, we derive from equation (54) a reciprocity relation for the Green’s
function $\check{\cal G}({\bf k},{\bf x}_{A},t,t_{A})$. Table 2 lists the
quantities for states $A$ and $B$ that we use in this derivation. The material
parameters are chosen the same in both states. For the wave field
$\check{U}_{A}$ we choose the acausal Green’s function $\check{\cal
G}^{a}({\bf k},{\bf x}_{A},t,t_{A})$. It obeys the same wave equation as the
causal Green’s function (equation (43)), but with the condition $\check{\cal
G}^{a}({\bf k},{\bf x}_{A},t,t_{A})=0$ for $t>t_{A}$. For the wave field
$\check{U}_{B}$ we choose the causal Green’s function $\check{\cal G}({\bf
k},{\bf x}_{B},t,t_{B})$, with $\check{\cal G}({\bf k},{\bf x}_{B},t,t_{B})=0$
for $t<t_{B}$. The expressions in Table 2 for the wave fields $\check{V}_{{\rm
d},A}$ and $\check{V}_{{\rm d},B}$ follow from equations (39) and (52) for
states $A$ and $B$ (using $\check{a}_{A}=\check{a}_{B}=0$). The expressions
for the source terms $\check{b}_{{\rm d},A}$ and $\check{b}_{{\rm d},B}$
follow from comparing the source term for a Green’s function (equation (43))
with that for a general wave field (equations (42) and (53), using
$\check{a}_{A}=\check{a}_{B}=0$). Note that the source term for the acausal
Green’s function in state $A$ actually represents a sink.

Table 2:  Quantities for deriving the reciprocity relation of equation (56). | State $A$ | State $B$
---|---|---
Material parameters | $\alpha_{A}=\alpha_{A}(t)$ | $\alpha_{B}=\alpha_{A}(t)$
| $\beta_{A}=\beta_{A}(t)$ | $\beta_{B}=\beta_{A}(t)$
Wave fields | $\check{U}_{A}=\check{\cal G}^{a}({\bf k},{\bf x}_{A},t,t_{A})$ | $\check{U}_{B}=\check{\cal G}({\bf k},{\bf x}_{B},t,t_{B})$
| $\check{V}_{{\rm d},A}=-\beta_{A}(t)\partial_{t}\check{\cal G}^{a}({\bf k},{\bf x}_{A},t,t_{A})$ | $\check{V}_{{\rm d},B}=-\beta_{A}(t)\partial_{t}\check{\cal G}({\bf k},{\bf x}_{B},t,t_{B})$
Sources | $\check{a}_{A}=0$ | $\check{a}_{B}=0$
| $\check{b}_{{\rm d},A}=-\exp(-i{\bf k}\cdot{\bf x}_{A})\delta(t-t_{A})$ | $\check{b}_{{\rm d},B}=-\exp(-i{\bf k}\cdot{\bf x}_{B})\delta(t-t_{B})$

We substitute the quantities of Table 2 into the reciprocity theorem of the
space-convolution type, formulated by equation (54). Let us assume that the
time instants $t_{A}$ and $t_{B}$ lie both between $t_{b}$ and $t_{e}$. Then
the acausal Green’s function is zero at $t_{e}$ and the causal Green’s
function is zero at $t_{b}$. With this, the left-hand side of equation (54)
vanishes. From the remaining terms on the right-hand side of equation (54) we
obtain

$\displaystyle\exp(i{\bf k}\cdot{\bf x}_{A})\check{\cal G}^{a}({\bf k},{\bf
x}_{A},t_{B},t_{A})=\exp(i{\bf k}\cdot{\bf x}_{B})\check{\cal G}({\bf k},{\bf
x}_{B},t_{A},t_{B}).$ (56)

This is the sought reciprocity relation for the Green’s function in the ${\bf
k},t$-domain. We use the inverse Fourier transformation defined in equation
(37) to transform this expression to the ${\bf x},t$-domain. This yields

$\displaystyle{\cal G}^{a}({\bf x}+{\bf x}_{A},{\bf x}_{A},t_{B},t_{A})={\cal
G}({\bf x}+{\bf x}_{B},{\bf x}_{B},t_{A},t_{B}),$ (57)

or, since in a homogeneous medium the Green’s function is shift invariant,

$\displaystyle{\cal G}^{a}({\bf x},{\bf 0},t_{B},t_{A})={\cal G}({\bf x},{\bf
0},t_{A},t_{B}).$ (58)

This is the counterpart of the well-known source-receiver reciprocity relation
${\cal G}({\bf x}_{B},{\bf x}_{A},t,0)={\cal G}({\bf x}_{A},{\bf x}_{B},t,0)$
for an inhomogeneous time-independent medium, which states that the source and
receiver positions can be interchanged (de Hoop, 1995; Morse & Feshbach,
1953). Equation (58) states that for a homogeneous time-dependent medium the
source and receiver times can be interchanged if, simultaneously, the causal
response is replaced by the acausal response. In the derivation we assumed
that $t_{A}$ and $t_{B}$ lie both between $t_{b}$ and $t_{e}$. If we take
$t_{b}\to-\infty$ and $t_{e}\to\infty$, then equations (56) – (58) hold for
arbitrary $t_{A}$ and $t_{B}$. Note that for $t_{A}<t_{B}$ these expressions
reduce to the trivial relation $0=0$.

###  7.3 Field-momentum density balance

The next special case we consider is the field-momentum density balance, which
we derive from equation (55). We take identical states $A$ and $B$ and drop
the subscripts. We thus obtain from equation (55)

$\displaystyle(\check{U}^{*}\check{V}_{\rm d}-\check{V}_{\rm
d}^{*}\check{U})\Bigr{|}_{t_{b}}^{t_{e}}=\int_{t_{b}}^{t_{e}}\bigl{(}\check{a}^{*}\check{V}_{\rm
d}-\check{b}_{\rm d}^{*}\check{U}-\check{V}_{\rm
d}^{*}\check{a}+\check{U}^{*}\check{b}_{\rm d}\bigr{)}{\rm d}t,$ (59)

or, using equations (47) and (52) and the definition ${\bf n}={\bf k}/|{\bf
k}|$,

$\displaystyle\check{M}({\bf k},t)\Bigr{|}_{t_{b}}^{t_{e}}$ $\displaystyle=$
$\displaystyle\Re\int_{t_{b}}^{t_{e}}\bigl{(}\check{a}^{*}\check{\bf
V}+\check{\bf b}^{*}\check{U}\bigr{)}\cdot{\bf n}{\rm d}t.$ (60)

This is the net field-momentum density balance for a homogeneous time-
dependent material for the situation that there are sources between $t_{b}$
and $t_{e}$. It is the counterpart of the net powerflux density balance for an
inhomogeneous time-independent medium (de Hoop, 1995; Fokkema & van den Berg,
1993).

##  8 Wave field representations

###  8.1 General wave field representation

A wave field representation is obtained by replacing one of the states in a
reciprocity theorem by a Green’s state (Knopoff, 1956; de Hoop, 1958; Gangi,
1970; Pao & Varatharajulu, 1976). We derive a general wave field
representation for a homogeneous time-dependent material with piecewise
continuous parameters $\alpha(t)$ and $\beta(t)$. To this end, consider the
quantities listed in Table 3. Note that state $A$ is the same as that in Table
2, except that ${\bf x}_{A}$ and $t_{A}$ are for convenience replaced by ${\bf
0}$ and $t^{\prime}$, respectively. State $B$ is the actual field in the
actual medium, so the subscripts $B$ are dropped.

Table 3:  Quantities for deriving the representation of equation (61). | State $A$ | State $B$
---|---|---
Material parameters | $\alpha_{A}=\alpha_{A}(t)$ | $\alpha_{B}=\alpha(t)$
| $\beta_{A}=\beta_{A}(t)$ | $\beta_{B}=\beta(t)$
Wave fields | $\check{U}_{A}=\check{\cal G}^{a}({\bf k},{\bf 0},t,t^{\prime})$ | $\check{U}_{B}=\check{U}({\bf k},t)$
| $\check{V}_{{\rm d},A}=-\beta_{A}(t)\partial_{t}\check{\cal G}^{a}({\bf k},{\bf 0},t,t^{\prime})$ | $\check{V}_{{\rm d},B}=\check{V}_{\rm d}({\bf k},t)$
Sources | $\check{a}_{A}=0$ | $\check{a}_{B}=\check{a}({\bf k},t)$
| $\check{b}_{{\rm d},A}=-\delta(t-t^{\prime})$ | $\check{b}_{{\rm d},B}=\check{b}_{\rm d}({\bf k},t)$

Substitution of the quantities in Table 3 into the reciprocity theorem of the
space-convolution type, formulated by equation (54), using reciprocity
relation (56) for the Green’s function with ${\bf x}_{A}={\bf x}_{B}={\bf 0}$,
yields

$\displaystyle\chi(t^{\prime})\check{U}({\bf
k},t^{\prime})=\Bigl{(}\check{\cal G}({\bf k},{\bf
0},t^{\prime},t)\check{V}_{\rm d}({\bf
k},t)+\beta_{A}(t)\\{\partial_{t}\check{\cal G}({\bf k},{\bf
0},t^{\prime},t)\\}\check{U}({\bf k},t)\Bigr{)}\Bigr{|}_{t=t_{b}}^{t=t_{e}}$
$\displaystyle-\int_{t_{b}}^{t_{e}}\Bigl{(}|{\bf k}|^{2}\check{\cal G}({\bf
k},{\bf 0},t^{\prime},t)\Delta\alpha^{-1}(t)\check{U}({\bf
k},t)-\beta_{A}(t)\\{\partial_{t}\check{\cal G}({\bf k},{\bf
0},t^{\prime},t)\\}\Delta\beta^{-1}(t)\check{V}_{\rm d}({\bf k},t)\Bigr{)}{\rm
d}t$ $\displaystyle-\int_{t_{b}}^{t_{e}}\Bigl{(}\\{\partial_{t}\check{\cal
G}({\bf k},{\bf 0},t^{\prime},t)\\}\beta_{A}(t)\check{a}({\bf
k},t)+\check{\cal G}({\bf k},{\bf 0},t^{\prime},t)\check{b}_{\rm d}({\bf
k},t)\Bigr{)}{\rm d}t,$ (61)

with the characteristic function $\chi(t^{\prime})$ defined as

$\displaystyle\chi(t^{\prime})=\begin{cases}&1\quad\mbox{for}\quad
t_{b}<t^{\prime}<t_{e},\\\ &\frac{1}{2}\quad\mbox{for}\quad
t^{\prime}=t_{b}\quad\mbox{or}\quad t^{\prime}=t_{e},\\\
&0\quad\mbox{for}\quad t^{\prime}<t_{b}\quad\mbox{or}\quad
t^{\prime}>t_{e},\end{cases}$ (62)

and with the material contrast parameters $\Delta\alpha^{-1}(t)$ and
$\Delta\beta^{-1}(t)$ defined as

$\displaystyle\Delta\alpha^{-1}(t)$ $\displaystyle=$
$\displaystyle\alpha^{-1}(t)-\alpha_{A}^{-1}(t),$ (63)
$\displaystyle\Delta\beta^{-1}(t)$ $\displaystyle=$
$\displaystyle\beta^{-1}(t)-\beta_{A}^{-1}(t).$ (64)

Equation (61) is the representation of the space-convolution type for a
homogeneous time-dependent medium in the ${\bf k},t$-domain. It is the
counterpart of the representation of the time-convolution type for an
inhomogeneous time-independent medium (Knopoff, 1956; de Hoop, 1958; Gangi,
1970; Pao & Varatharajulu, 1976). The left-hand side represents the wave field
in the actual medium at time instant $t^{\prime}$. The first term on the
right-hand side describes the contributions from the fields at $t_{b}$ and
$t_{e}$ (when $t^{\prime}$ is between $t_{b}$ and $t_{e}$ then only the field
at $t_{b}$ contributes). This term is the counterpart of the spatial
Kirchhoff-Helmholtz integral (Morse & Feshbach, 1953; Berkhout, 1985; Frazer &
Sen, 1985; Druzhinin et al., 1998; Tygel et al., 2000), which is an integral
along an enclosing boundary (since time is one-dimensional, here the
“enclosing boundary” only consists of the time instants $t_{b}$ and $t_{e}$).
If there were no contrasts of material parameters and no sources between
$t_{b}$ and $t_{e}$, then this first term would give the complete solution
(similar as the Kirchhoff-Helmholtz integral would give the complete solution
if there were no contrasts and no sources inside the enclosing boundary). The
second term on the right-hand side accounts for contrasts between the material
parameters of the actual medium and those of the medium in which the Green’s
function is defined. Hence, this integral describes wave scattering at
parameter contrasts in time-dependent materials. The third term on the right-
hand side accounts for the contribution of sources between $t_{b}$ and
$t_{e}$.

We transform equation (61) to the ${\bf x},t$-domain, using the inverse
Fourier transformation defined in equation (37). Using equations (52) and (53)
and the shift-invariance of the Green’s functions, we obtain

$\displaystyle\chi(t^{\prime})U({\bf
x}^{\prime},t^{\prime})=\int_{\mathbb{R}^{2}}\Bigl{(}{\cal G}({\bf
x}^{\prime},{\bf x},t^{\prime},t){\bf\nabla}\cdot{\bf V}({\bf
x},t)+\beta_{A}(t)\\{\partial_{t}{\cal G}({\bf x}^{\prime},{\bf
x},t^{\prime},t)\\}U({\bf x},t)\Bigr{)}{\rm d}{\bf
x}\Bigr{|}_{t=t_{b}}^{t=t_{e}}$
$\displaystyle+\int_{t_{b}}^{t_{e}}\int_{\mathbb{R}^{2}}\Bigl{(}{\cal G}({\bf
x}^{\prime},{\bf x},t^{\prime},t)\Delta\alpha^{-1}(t){\bf\nabla}^{2}U({\bf
x},t)+\beta_{A}(t)\\{\partial_{t}{\cal G}({\bf x}^{\prime},{\bf
x},t^{\prime},t)\\}\Delta\beta^{-1}(t){\bf\nabla}\cdot{\bf V}({\bf
x},t)\Bigr{)}{\rm d}{\bf x}{\rm d}t$
$\displaystyle-\int_{t_{b}}^{t_{e}}\int_{\mathbb{R}^{2}}\Bigl{(}\\{\partial_{t}{\cal
G}({\bf x}^{\prime},{\bf x},t^{\prime},t)\\}\beta_{A}(t)a({\bf x},t)+{\cal
G}({\bf x}^{\prime},{\bf x},t^{\prime},t){\bf\nabla}\cdot{\bf b}({\bf
x},t)\Bigr{)}{\rm d}{\bf x}{\rm d}t.$ (65)

Equation (65) is the representation of the space-convolution type for a
homogeneous time-dependent medium in the ${\bf x},t$-domain. Note that
equation (33), with the source function $s$ defined in equation (11), is
obtained as a special case of this representation if we choose the same
material parameters in both states, replace $t_{b}$ and $t_{e}$ by $-\infty$
and $\infty$, respectively, and apply integration by parts to replace
$-\int(\partial_{t}{\cal G})\beta a{\rm d}t$ by $\int{\cal
G}\partial_{t}(\beta a){\rm d}t$. In its general form, equation (65) provides
a basis for the analysis of wave scattering problems in homogeneous time-
dependent media. We discuss an example in the next subsection.

###  8.2 Wave field representation for the case of a single time boundary

We use the representation of equation (65) to analyze the wave field after a
time boundary between two time-independent slabs, as illustrated in the
numerical example in section 4. To this end we take $t_{b}$ equal to the time
boundary (i.e., $t_{b}=50\,\mu$s) and $t_{e}\to\infty$. Hence, the boundary
$t=t_{e}$ in the first term on the right-hand side of equation (65) gives no
contribution. The wave field $U({\bf x},t)$ is the response to a spatial
source distribution $s_{0}({\bf x})$ at $t=0$ in a medium with velocity
$c_{1}=1500$ m/s for $t<t_{b}$ and velocity $c_{2}=2500$ m/s for $t>t_{b}$.
The parameter $\beta$ is constant in the numerical example, but here we assume
it has the values $\beta_{1}$ and $\beta_{2}$ before and after the time
boundary, respectively (and consequently $\alpha_{1}=1/\beta_{1}c_{1}^{2}$ and
$\alpha_{2}=1/\beta_{2}c_{2}^{2}$ before and after the time boundary). We
define the Green’s function in equation (65) in a time-independent medium with
velocity $c_{A}=c_{2}$ and parameter $\beta_{A}=\beta_{2}$ (and
$\alpha_{A}=1/\beta_{2}c_{2}^{2}$) and call this ${\cal G}_{2}({\bf
x}^{\prime},{\bf x},t^{\prime},t)$. Note that for all $t>t_{b}$ we thus have
$\Delta\alpha^{-1}(t)=\Delta\beta^{-1}(t)=0$, hence, the second term on the
right-hand side of equation (65) vanishes. Since the source distribution for
$U({\bf x},t)$ exists only at $t=0<t_{b}$, the third term on the right-hand
side of equation (65) also vanishes. Hence, for $t^{\prime}>t_{b}$ we are left
with

$\displaystyle U({\bf
x}^{\prime},t^{\prime})=-\int_{\mathbb{R}^{2}}\Bigl{[}{\cal G}_{2}({\bf
x}^{\prime},{\bf x},t^{\prime},t){\bf\nabla}\cdot{\bf V}({\bf
x},t)+\beta_{2}\\{\partial_{t}{\cal G}_{2}({\bf x}^{\prime},{\bf
x},t^{\prime},t)\\}U({\bf x},t)\Bigr{]}_{t=t_{b}}{\rm d}{\bf x}.$ (66)

In the ${\bf k},t$-domain this becomes

$\displaystyle\check{U}({\bf k},t^{\prime})=-\Bigl{[}\check{\cal G}({\bf
k},{\bf 0},t^{\prime},t)\check{V}_{\rm d}({\bf
k},t)+\beta_{2}\\{\partial_{t}\check{\cal G}_{2}({\bf k},{\bf
0},t^{\prime},t)\\}\check{U}({\bf k},t)\Bigr{]}_{t=t_{b}},\quad\mbox{for}\quad
t^{\prime}>t_{b}.$ (67)

For $t=t_{b}$ and $t^{\prime}>t_{b}$ we have, according to equation (45),

$\displaystyle\check{\cal G}_{2}({\bf k},{\bf 0},t^{\prime},t_{b})$
$\displaystyle=$ $\displaystyle\frac{\sin(|{\bf
k}|c_{2}(t^{\prime}-t_{b}))}{\eta_{2}|{\bf k}|},$ (68)
$\displaystyle\beta_{2}\partial_{t}\check{\cal G}_{2}({\bf k},{\bf
0},t^{\prime},t)|_{t=t_{b}}$ $\displaystyle=$ $\displaystyle-\cos(|{\bf
k}|c_{2}(t^{\prime}-t_{b})),$ (69)

with $\eta_{2}=\beta_{2}c_{2}$. From equations (34) and (36), we further
obtain for $t=t_{b}$

$\displaystyle\check{U}({\bf k},t_{b})$ $\displaystyle=$
$\displaystyle\frac{\sin(|{\bf k}|c_{1}t_{b})}{\eta_{1}|{\bf
k}|}\check{s}_{0}({\bf k}),$ (70)

with $\eta_{1}=\beta_{1}c_{1}$ and, using equations (52) and (39) with
$\check{a}({\bf k},t_{b})=0$,

$\displaystyle\check{V}_{\rm d}({\bf k},t_{b})$ $\displaystyle=$
$\displaystyle-\cos(|{\bf k}|c_{1}t_{b})\check{s}_{0}({\bf k}).$ (71)

Substituting these expressions into equation (67) yields

$\displaystyle\check{U}({\bf
k},t^{\prime})=\biggl{(}\frac{\sin\gamma_{1}\cos\gamma_{2}}{\eta_{1}|{\bf
k}|}+\frac{\cos\gamma_{1}\sin\gamma_{2}}{\eta_{2}|{\bf
k}|}\biggr{)}\check{s}_{0}({\bf k}),\quad\mbox{for}\quad t^{\prime}>t_{b},$
(72)

with

$\displaystyle\gamma_{1}$ $\displaystyle=$ $\displaystyle|{\bf k}|c_{1}t_{b},$
(73) $\displaystyle\gamma_{2}$ $\displaystyle=$ $\displaystyle|{\bf
k}|c_{2}(t^{\prime}-t_{b}).$ (74)

Equation (72) can be rewritten as

$\displaystyle\check{U}({\bf
k},t^{\prime})=\biggl{(}T_{u}\frac{\sin(\gamma_{1}+\gamma_{2})}{\eta_{1}|{\bf
k}|}+R_{u}\frac{\sin(\gamma_{1}-\gamma_{2})}{\eta_{1}|{\bf
k}|}\biggr{)}\check{s}_{0}({\bf k}),\quad\mbox{for}\quad t^{\prime}>t_{b},$
(75)

with $T_{u}$ and $R_{u}$ defined in equations (23) and (24), respectively
(this is verified by substituting
$\sin(\gamma_{1}\pm\gamma_{2})=\sin\gamma_{1}\cos\gamma_{2}\pm\cos\gamma_{1}\sin\gamma_{2}$
into equation (75), and using equations (21) and (22)). Let us write

$\displaystyle\gamma_{1}+\gamma_{2}$ $\displaystyle=$ $\displaystyle|{\bf
k}|c_{2}(t^{\prime}-t_{V}),\quad\mbox{with}\quad
t_{V}=\bigl{(}1-\frac{c_{1}}{c_{2}}\bigr{)}t_{b},$ (76)
$\displaystyle\gamma_{1}-\gamma_{2}$ $\displaystyle=$ $\displaystyle|{\bf
k}|c_{2}(t_{F}-t^{\prime}),\quad\mbox{with}\quad
t_{F}=\bigl{(}1+\frac{c_{1}}{c_{2}}\bigr{)}t_{b}.$ (77)

First, consider equation (76). Since $t^{\prime}>t_{b}$ and $t_{V}<t_{b}$, we
have $t^{\prime}>t_{V}$ and, consequently, $\gamma_{1}+\gamma_{2}>0$. Hence,
for the first term between the large brackets in equation (75) we find, using
equation (45),

$\displaystyle T_{u}\frac{\sin(\gamma_{1}+\gamma_{2})}{\eta_{1}|{\bf
k}|}=\frac{\eta_{2}}{\eta_{1}}T_{u}\frac{\sin(|{\bf
k}|c_{2}(t^{\prime}-t_{V}))}{\eta_{2}|{\bf
k}|}=\frac{\eta_{2}}{\eta_{1}}T_{u}\check{\cal G}_{2}({\bf k},{\bf
0},t^{\prime},t_{V}),\quad\mbox{for}\quad t^{\prime}>t_{b}.$ (78)

Next, consider equation (77). Since $t^{\prime}>t_{b}$ and $t_{F}>t_{b}$, the
term $\gamma_{1}-\gamma_{2}$ can take positive as well as negative values. For
the second term between the large brackets in equation (75) we find, using
equations (45) and (56),

$\displaystyle R_{u}\frac{\sin(\gamma_{1}-\gamma_{2})}{\eta_{1}|{\bf k}|}$
$\displaystyle=$
$\displaystyle\frac{\eta_{2}}{\eta_{1}}R_{u}\Bigl{(}H(t_{F}-t^{\prime})+H(t^{\prime}-t_{F})\Bigr{)}\frac{\sin(|{\bf
k}|c_{2}(t_{F}-t^{\prime}))}{\eta_{2}|{\bf k}|}$ (79) $\displaystyle=$
$\displaystyle\frac{\eta_{2}}{\eta_{1}}R_{u}\Bigl{(}\check{\cal
G}_{2}^{a}({\bf k},{\bf 0},t^{\prime},t_{F})-\check{\cal G}_{2}({\bf k},{\bf
0},t^{\prime},t_{F})\Bigr{)},\quad\mbox{for}\quad t^{\prime}>t_{b}.$

Substituting equations (78) and (79) into equation (75) and applying an
inverse spatial Fourier transform yields

$\displaystyle U({\bf
x}^{\prime},t^{\prime})=\frac{\eta_{2}}{\eta_{1}}\int_{\mathbb{R}^{2}}\biggl{[}T_{u}{\cal
G}_{2}({\bf x}^{\prime},{\bf x},t^{\prime},t_{V})+R_{u}\Bigl{(}{\cal
G}_{2}^{a}({\bf x}^{\prime},{\bf x},t^{\prime},t_{F})-{\cal G}_{2}({\bf
x}^{\prime},{\bf x},t^{\prime},t_{F})\Bigr{)}\biggr{]}s_{0}({\bf x}){\rm
d}{\bf x},$ (80)

for $t^{\prime}>t_{b}$. The first Green’s function on the right-hand side
describes the transmitted, outward propagating wave in Figures 1 and 2 for
$t^{\prime}>t_{b}=50\,\mu$s in a material with propagation velocity $c_{2}$
and parameter $\beta_{2}$. It originates from a virtual source at time instant
$t_{V}$, defined in equation (76), hence, at $t_{V}=20\,\mu$s. The amplitude
of this transmitted wave is proportional to the transmission coefficient
$T_{u}=0.8$ and a factor $\eta_{2}/\eta_{1}$ to account for the fact that the
actual source is situated in a material with propagation velocity $c_{1}$ and
parameter $\beta_{1}$. The second Green’s function is acausal and describes
the reflected, inward propagating wave in Figures 1 and 2 for
$t_{b}<t^{\prime}<t_{F}$. It focuses at time instant $t_{F}$, defined in
equation (77), hence, at $t_{F}=80\,\mu$s. The amplitude of this reflected
wave is proportional to the reflection coefficient $R_{u}=0.2$ and the factor
$\eta_{2}/\eta_{1}$. The third Green’s function is causal and describes the
continuation of the reflected wave in Figures 1 and 2 for
$t^{\prime}>t_{F}=80\,\mu$s, i.e., after focusing. It originates from a
virtual source at the focal time $t_{F}=80\,\mu$s. The reflected wave
undergoes a sign change at the focal time, hence, its amplitude after focusing
is proportional to $-R_{u}=-0.2$ and the factor $\eta_{2}/\eta_{1}$. The
integration over ${\bf x}$ in equation (80) describes the spatial convolution
of the superposition of these three Green’s functions with the spatial source
distribution $s_{0}({\bf x})$ at $t=0$. This explains Figure 3 for
$t^{\prime}>t_{b}=50\,\mu$s, and in particular the recovery of the source
distribution $s_{0}({\bf x})$ around the focal time $t_{F}=80\,\mu$s, where
${\cal G}_{2}^{a}({\bf x}^{\prime},{\bf x},t^{\prime},t_{F})$ focuses.

##  9 Green’s function retrieval

For inhomogeneous time-independent media, it has been shown that, under
specific circumstances, the time-correlation of passive wave measurements at
two receivers converges to the response at one of these receivers as if there
were an impulsive source at the position of the other (Weaver & Lobkis, 2001;
Campillo & Paul, 2003; Wapenaar, 2003; Snieder, 2004; Malcolm et al., 2004;
Roux et al., 2004; Haney, 2009; Sabra et al., 2007). In other words, the
Green’s function between two receivers is retrieved by time-correlating
passive responses at these receivers. Using a reciprocity theorem of the time-
correlation type (Wapenaar & Fokkema, 2006), it has been shown that this holds
for arbitrary inhomogeneous time-independent media. Here we use the
reciprocity theorem of the space-correlation type of equation (55) to derive
an expression for Green’s function retrieval in a homogeneous time-dependent
material with piecewise continuous parameters $\alpha(t)$ and $\beta(t)$. To
this end, we start with acausal Green’s functions in both states, with unit
sinks at $t_{A}$ and $t_{B}$, both between $t_{b}$ and $t_{e}$ (here $t_{b}$
and $t_{e}$ are again arbitrary and not related to a time boundary, such as
$t_{b}$ in section 8.2). Substituting the quantities of Table 4 into the
reciprocity theorem of the space-correlation type (equation (55)), using
reciprocity relation (56) for the Green’s functions with ${\bf x}_{A}={\bf
x}_{B}={\bf 0}$, yields

Table 4:  Quantities for deriving the expression of equation (81) for Green’s function retrieval. | State $A$ | State $B$
---|---|---
Material parameters | $\alpha_{A}=\alpha(t)$ | $\alpha_{B}=\alpha(t)$
| $\beta_{A}=\beta(t)$ | $\beta_{B}=\beta(t)$
Wave fields | $\check{U}_{A}=\check{\cal G}^{a}({\bf k},{\bf 0},t,t_{A})$ | $\check{U}_{B}=\check{\cal G}^{a}({\bf k},{\bf 0},t,t_{B})$
| $\check{V}_{{\rm d},A}=-\beta(t)\partial_{t}\check{\cal G}^{a}({\bf k},{\bf 0},t,t_{A})$ | $\check{V}_{{\rm d},B}=-\beta(t)\partial_{t}\check{\cal G}^{a}({\bf k},{\bf 0},t,t_{B})$
Sources | $\check{a}_{A}=0$ | $\check{a}_{B}=0$
| $\check{b}_{{\rm d},A}=-\delta(t-t_{A})$ | $\check{b}_{{\rm d},B}=-\delta(t-t_{B})$

$\displaystyle\check{\cal G}({\bf k},{\bf 0},t_{B},t_{A})-\\{\check{\cal
G}^{a}({\bf k},{\bf 0},t_{B},t_{A})\\}^{*}=$
$\displaystyle\beta(t_{b})\Bigl{[}\check{\cal G}^{*}({\bf k},{\bf
0},t_{A},t)\partial_{t}\check{\cal G}({\bf k},{\bf
0},t_{B},t)-\\{\partial_{t}\check{\cal G}^{*}({\bf k},{\bf
0},t_{A},t)\\}\check{\cal G}({\bf k},{\bf 0},t_{B},t)\Bigr{]}_{t=t_{b}}.$ (81)

Applying the inverse Fourier transform of equation (37) to both sides of this
equation yields

$\displaystyle{\cal G}({\bf x}^{\prime},{\bf 0},t_{B},t_{A})-{\cal
G}^{a}(-{\bf x}^{\prime},{\bf 0},t_{B},t_{A})=$
$\displaystyle\beta(t_{b})\int_{\mathbb{R}^{2}}\Bigl{[}{\cal G}({\bf x},{\bf
0},t_{A},t)\partial_{t}{\cal G}({\bf x}^{\prime}+{\bf x},{\bf
0},t_{B},t)-\\{\partial_{t}{\cal G}({\bf x},{\bf 0},t_{A},t)\\}{\cal G}({\bf
x}^{\prime}+{\bf x},{\bf 0},t_{B},t)\Bigr{]}_{t=t_{b}}{\rm d}{\bf x}.$ (82)

The right-hand side consists of space correlations of responses to an
impulsive source at $t_{b}$ and its derivative, observed by receivers at
$t_{A}$ and $t_{B}$. The left-hand side consists of a Green’s function and its
acausal version for a source or sink at $t_{A}$, observed at $t_{B}$. When
$t_{B}$ is larger than $t_{A}$ the causal Green’s function ${\cal G}({\bf
x}^{\prime},{\bf 0},t_{B},t_{A})$ is retrieved (since the acausal Green’s
function vanishes for this situation). On the other hand, the acausal Green’s
function $-{\cal G}^{a}(-{\bf x}^{\prime},{\bf 0},t_{B},t_{A})$ is retrieved
when $t_{B}$ is smaller than $t_{A}$. Note that equation (82) holds for any
time instant $t_{b}$ of the original source, as long $t_{A}$ and $t_{B}$ are
both larger than $t_{b}$. Contrary to Green’s function retrieval by time
correlation in space-dependent media, which requires sources on a boundary
enclosing the receivers, for Green’s function retrieval by space correlation
in time-dependent media it suffices to have a single source at $t_{b}$, prior
to the receivers at $t_{A}$ and $t_{B}$.

We confirm equation (81) for the situation of a time-independent medium.
Assuming $t_{A}>t_{b}$ and $t_{B}>t_{b}$, the Green’s functions on the right-
hand side for a time-independent medium read, according to equation (45),

$\displaystyle\check{\cal G}({\bf k},{\bf 0},t_{A},t_{b})$ $\displaystyle=$
$\displaystyle\frac{\sin(|{\bf k}|c\Delta t_{A})}{\eta|{\bf k}|},$ (83)
$\displaystyle\partial_{t}\check{\cal G}({\bf k},{\bf 0},t_{B},t)|_{t=t_{b}}$
$\displaystyle=$ $\displaystyle-\frac{1}{\beta}\cos(|{\bf k}|c\Delta t_{B})$
(84)

and similar expressions for the other Green’s functions, with

$\displaystyle\Delta t_{A}$ $\displaystyle=$ $\displaystyle t_{A}-t_{b},$ (85)
$\displaystyle\Delta t_{B}$ $\displaystyle=$ $\displaystyle t_{B}-t_{b}.$ (86)

Hence, for the right-hand side of equation (81) we obtain

$\displaystyle-\frac{\sin(|{\bf k}|c\Delta t_{A})\cos(|{\bf k}|c\Delta
t_{B})-\cos(|{\bf k}|c\Delta t_{A})\sin(|{\bf k}|c\Delta t_{B})}{\eta|{\bf
k}|}=\frac{\sin\bigl{(}|{\bf k}|c(t_{B}-t_{A})\bigr{)}}{\eta|{\bf k}|}.$ (87)

For $t_{B}>t_{A}$ this is equal to $\check{\cal G}({\bf k},{\bf
0},t_{B},t_{A})$, which is the left-hand side of equation (81), since
$\check{\cal G}^{a}({\bf k},{\bf 0},t_{B},t_{A})=0$ for this situation. On the
other hand, for $t_{B}<t_{A}$ it is equal to $-\check{\cal G}^{a}({\bf k},{\bf
0},t_{B},t_{A})$. This is again the left-hand side of equation (81), since
$\check{\cal G}^{a}({\bf k},{\bf 0},t_{B},t_{A})$ is real-valued and
$\check{\cal G}({\bf k},{\bf 0},t_{B},t_{A})=0$ for this situation.

##  10 Matrix-vector wave equation

Many authors use a matrix-vector formalism to conveniently analyse wave
propagation and scattering in time-dependent media (Salem & Caloz, 2015;
Torrent et al., 2018; Pacheco-Peña & Engheta, 2020; Caloz & Deck-Léger, 2020a;
Wapenaar et al., 2024). Here we recast equations (7) and (8) into a matrix-
vector wave equation in the ${\bf x},t$-domain. Unlike for the 1D situation,
where equations similar to (7) and (8) govern the scalar wave fields $U$ and
$V$ (Wapenaar et al., 2024), here these equations govern $U$ and ${\bf V}$,
where ${\bf V}$ is a vectorial quantity. Analogous to equation (52), we define
a scalar wave field $V_{\rm d}$ as

$\displaystyle V_{\rm d}$ $\displaystyle=$ $\displaystyle{\bf\nabla}\cdot{\bf
V},$ (88)

where subscript ${\rm d}$ refers to “divergence”. Using this in equation (7)
and applying the divergence operator to both sides of equation (8) yields

$\displaystyle\partial_{t}U+\frac{1}{\beta}V_{\rm d}$ $\displaystyle=$
$\displaystyle a,$ (89) $\displaystyle\partial_{t}V_{\rm
d}+\frac{1}{\alpha}{\bf\nabla}^{2}U$ $\displaystyle=$ $\displaystyle b_{\rm
d},$ (90)

where, analogous to equation (53),

$\displaystyle b_{\rm d}$ $\displaystyle=$ $\displaystyle{\bf\nabla}\cdot{\bf
b}.$ (91)

Equations (89) and (90) for the scalar fields $U$ and $V_{\rm d}$ can be
combined into the following matrix-vector wave equation

$\displaystyle\partial_{t}{\bf q}_{\rm d}={\bf A}_{\rm d}{\bf q}_{\rm d}+{\bf
d}_{\rm d},$ (92)

with wave field vector ${\bf q}_{\rm d}({\bf x},t)$, matrix ${\bf A}_{\rm
d}({\bf x},t)$ and source vector ${\bf d}_{\rm d}({\bf x},t)$ defined as

$\displaystyle{\bf q}_{\rm d}=\begin{pmatrix}U\\\ V_{\rm
d}\end{pmatrix},\quad{\bf A}_{\rm d}=\begin{pmatrix}0&-\frac{1}{\beta}\\\
-\frac{1}{\alpha}{\bf\nabla}^{2}&0\end{pmatrix},\quad{\bf d}_{\rm
d}=\begin{pmatrix}a\\\ b_{\rm d}\end{pmatrix}.$ (93)

The subscript ${\rm d}$ in ${\bf q}_{\rm d}$, ${\bf A}_{\rm d}$ and ${\bf
d}_{\rm d}$ is used to distinguish these quantities from those in the 1D
situation where, for example, wave field vector ${\bf q}$ contains the scalar
wave fields $U$ and $V$ (Wapenaar et al., 2024). Equation (92) holds for a
homogeneous time-dependent medium with continuous parameters $\alpha(t)$ and
$\beta(t)$. For a medium with piecewise continuous parameters, ${\bf q}_{\rm
d}({\bf x},t)$ remains continuous at those time instants where $\alpha(t)$ and
$\beta(t)$ are discontinuous. Equation (92) is the counterpart of a matrix-
vector wave equation for an inhomogeneous time-independent material (Corones,
1975; Kosloff & Baysal, 1983; Fishman & McCoy, 1984; Ursin, 1983; Wapenaar &
Berkhout, 1986; Løseth & Ursin, 2007).

Applying the 2D Fourier transformation defined in equation (36) to equation
(92), we obtain the following matrix-vector wave equation in the ${\bf
k},t$-domain

$\displaystyle\partial_{t}\check{\bf q}_{\rm d}=\check{\bf A}_{\rm
d}\check{\bf q}_{\rm d}+\check{\bf d}_{\rm d},$ (94)

with wave field vector $\check{\bf q}_{\rm d}({\bf k},t)$, matrix $\check{\bf
A}_{\rm d}({\bf k},t)$ and source vector $\check{\bf d}_{\rm d}({\bf k},t)$
defined as

$\displaystyle\check{\bf q}_{\rm d}=\begin{pmatrix}\check{U}\\\ \check{V}_{\rm
d}\end{pmatrix},\quad\check{\bf A}_{\rm
d}=\begin{pmatrix}0&-\frac{1}{\beta}\\\ \frac{1}{\alpha}|{\bf
k}|^{2}&0\end{pmatrix},\quad\check{\bf d}_{\rm d}=\begin{pmatrix}\check{a}\\\
\check{b}_{\rm d}\end{pmatrix},$ (95)

with $\check{V}_{\rm d}$ and $\check{b}_{\rm d}$ defined in equations (52) and
(53), respectively. Note that $\check{\bf A}_{\rm d}$ obeys the symmetry
property

$\displaystyle\check{\bf A}_{\rm d}^{t}{\bf N}=-{\bf N}\check{\bf A}_{\rm d},$
(96)

where superscript $t$ denotes transposition and where

$\displaystyle{{\bf N}}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$ (97)

##  11 Propagator matrix

###  11.1 Propagator matrix in space-time domain

We define the propagator matrix ${\bf W}_{\rm d}({\bf x},t,t_{0})$ as the
solution of equation (92) without the source term, hence

$\displaystyle\partial_{t}{\bf W}_{\rm d}={\bf A}_{\rm d}{\bf W}_{\rm d},$
(98)

with initial condition

$\displaystyle{\bf W}_{\rm d}({\bf x},t_{0},t_{0})={\bf I}\delta({\bf x}),$
(99)

where ${\bf I}$ is the identity matrix (Torrent et al., 2018; Salem & Caloz,
2015; Pacheco-Peña & Engheta, 2020). We obtain a simple representation for
${\bf q}_{\rm d}({\bf x},t)$ in terms of ${\bf W}_{\rm d}({\bf x},t,t_{0})$,
assuming they both reside in the same medium and assuming there are no sources
for ${\bf q}_{\rm d}({\bf x},t)$ between $t_{0}$ and $t$. Whereas ${\bf
q}_{\rm d}({\bf x},t)$ can have any space-dependency at $t=t_{0}$, ${\bf
W}_{\rm d}({\bf x},t,t_{0})$ collapses to ${\bf I}\delta({\bf x})$ at
$t=t_{0}$. Hence, we obtain the following representation for ${\bf q}_{\rm
d}({\bf x},t)$ by applying Huygens’ superposition principle, according to

$\displaystyle{\bf q}_{\rm d}({\bf x},t)=\int_{\mathbb{R}^{2}}{\bf W}_{\rm
d}({\bf x}-{\bf x}_{0},t,t_{0}){\bf q}_{\rm d}({\bf x}_{0},t_{0}){\rm d}{\bf
x}_{0},$ (100)

where ${\bf x}_{0}$ is a variable. According to this equation, ${\bf W}_{\rm
d}({\bf x},t,t_{0})$ propagates the wave field vector ${\bf q}_{\rm d}$ from
$t_{0}$ to $t$ (where $t$ can be larger or smaller than $t_{0}$). The
subscript ${\rm d}$ in ${\bf W}_{\rm d}$ denotes that this propagator matrix
acts on a wave field vector containing $U$ and $V_{\rm d}$, which is different
from the 1D version of the propagator matrix, which acts on a wave field
vector containing $U$ and $V$ (Wapenaar et al., 2024). The propagator matrix
${\bf W}_{\rm d}({\bf x},t,t_{0})$ for a homogeneous time-dependent material
is the counterpart of a propagator matrix for an inhomogeneous time-
independent material (Thomson, 1950; Haskell, 1953; Gilbert & Backus, 1966;
Kennett, 1983, 1972; Woodhouse, 1974).

We partition ${\bf W}_{\rm d}({\bf x},t,t_{0})$ as follows

$\displaystyle{\bf W}_{\rm d}({\bf x},t,t_{0})=\begin{pmatrix}W_{\rm
d}^{U,U}&W_{\rm d}^{U,V}\\\ &\\\ W_{\rm d}^{V,U}&W_{\rm
d}^{V,V}\end{pmatrix}({\bf x},t,t_{0}).$ (101)

The first superscript refers to the wave field quantities in ${\bf q}_{\rm
d}({\bf x},t)$ in equation (100) and the second superscript to those in ${\bf
q}_{\rm d}({\bf x}_{0},t_{0})$ (with superscript $V$ referring to wave field
quantity $V_{\rm d}$).

By applying equation (100) recursively, we find the following recursive
expression for ${\bf W}_{\rm d}$

$\displaystyle{\bf W}_{\rm d}({\bf x},t_{N},t_{0})={\bf W}_{\rm d}({\bf
x},t_{N},t_{N-1})*_{x}\cdots*_{x}{\bf W}_{\rm d}({\bf
x},t_{n},t_{n-1})*_{x}\cdots*_{x}{\bf W}_{\rm d}({\bf x},t_{1},t_{0}),$ (102)

where $*_{x}$ denotes a 2D spatial convolution, similar as in equation (100),
and where $t_{1}\cdots t_{n}\cdots t_{N-1}$ are time instants where the medium
parameters $\alpha(t)$ and $\beta(t)$ may be discontinuous. Between these time
instants, the parameters $\alpha(t)$ and $\beta(t)$ can, in general, be
continuous functions of $t$. In Appendix B we give an explicit expression for
${\bf W}_{\rm d}({\bf x},t_{n},t_{n-1})$, assuming a time-independent slab
between $t_{n-1}$ and $t_{n}$.

As an illustration, we consider a piecewise constant material consisting of
five time-independent slabs, with propagation velocities of 2000, 2000, 1200,
2500 and 1400 m/s, respectively. The parameter $\beta$ is taken constant. The
duration of each time slab is 25 $\mu$s. We choose $t_{0}$ between the first
and second time slab, i.e., $t_{0}=25\,\mu$s. Figure 4(a) shows a
$z,t$-diagram of the propagator element $W_{\rm d}^{U,V}({\bf x},t,t_{0})$
(convolved with a spatial wavelet) for $x=0$ and $t_{0}=25\,\mu$s. The green
line at $t_{0}=25\,\mu$s indicates the initial condition of equation (99),
which for the considered off-diagonal element implies $W_{\rm d}^{U,V}({\bf
x},t_{0},t_{0})=0$. In the first time slab right of this green line, the
propagator consists (for $x=0$) of upward and downward propagating direct
waves. At each time boundary (indicated by the dashed blue lines) these waves
split into upward and downward propagating transmitted and reflected waves.
This is a manifestation of the recursive character of the propagator, as
formulated by equation (102).

Figure 4: Cross-sections at $x=0$ of (a) the propagator element $W_{\rm
d}^{U,V}({\bf x},t,t_{0})$ and (b) the Green’s function ${\cal G}({\bf x},{\bf
0},t,t_{0})$ (both convolved with a spatial wavelet) for a piecewise constant
time-dependent material. Movie of (a) available at
https://www.keeswapenaar.nl/TimeMaterial/Wuv.mp4.

###  11.2 Propagator matrix in wave-vector time domain

Applying the 2D Fourier transformation defined in equation (36) to equations
(98) – (100) we obtain

$\displaystyle\partial_{t}\check{\bf W}_{\rm d}=\check{\bf A}_{\rm
d}\check{\bf W}_{\rm d},$ (103)

with initial condition

$\displaystyle\check{\bf W}_{\rm d}({\bf k},t_{0},t_{0})={\bf I}$ (104)

and representation

$\displaystyle\check{\bf q}_{\rm d}({\bf k},t)=\check{\bf W}_{\rm d}({\bf
k},t,t_{0})\check{\bf q}_{\rm d}({\bf k},t_{0}).$ (105)

Analogous to equation (101) we partition $\check{\bf W}_{\rm d}({\bf
k},t,t_{0})$ as

$\displaystyle\check{\bf W}_{\rm d}({\bf
k},t,t_{0})=\begin{pmatrix}\check{W}_{\rm d}^{U,U}&\check{W}_{\rm d}^{U,V}\\\
&\\\ \check{W}_{\rm d}^{V,U}&\check{W}_{\rm d}^{V,V}\end{pmatrix}({\bf
k},t,t_{0}).$ (106)

By applying equation (105) recursively, we find the following recursive
expression for $\check{\bf W}_{\rm d}$

$\displaystyle\check{\bf W}_{\rm d}({\bf k},t_{N},t_{0})$ $\displaystyle=$
$\displaystyle\check{\bf W}_{\rm d}({\bf k},t_{N},t_{N-1})\cdots\check{\bf
W}_{\rm d}({\bf k},t_{n},t_{n-1})\cdots\check{\bf W}_{\rm d}({\bf
k},t_{1},t_{0}),$ (107)

where $t_{1}\cdots t_{n}\cdots t_{N-1}$ are time instants where the medium
parameters $\alpha(t)$ and $\beta(t)$ may be discontinuous. Between these time
instants, the parameters $\alpha(t)$ and $\beta(t)$ can, in general, be
continuous functions of $t$. In Appendix B we give an explicit expression for
$\check{\bf W}_{\rm d}({\bf k},t_{n},t_{n-1})$, assuming a time-independent
slab between $t_{n-1}$ and $t_{n}$.

###  11.3 Symmetry properties of the propagator matrix

To derive symmetry properties of the propagator matrix, we first show that the
quantity $\check{\bf W}_{\rm d}^{t}({\bf k},t,t_{A}){\bf N}\check{\bf W}_{\rm
d}({\bf k},t,t_{B})$ is a propagation invariant, i.e., that it is independent
of $t$. Taking the time derivative, applying the product rule for
differentiation, using equations (103) and (96), we obtain

$\displaystyle\partial_{t}\\{\check{\bf W}_{\rm d}^{t}({\bf k},t,t_{A}){\bf
N}\check{\bf W}_{\rm d}({\bf k},t,t_{B})\\}$ $\displaystyle\hskip
56.9055pt=\\{\partial_{t}\check{\bf W}_{\rm d}({\bf k},t,t_{A})\\}^{t}{\bf
N}\check{\bf W}_{\rm d}({\bf k},t,t_{B})+\check{\bf W}_{\rm d}^{t}({\bf
k},t,t_{A}){\bf N}\\{\partial_{t}\check{\bf W}_{\rm d}({\bf k},t,t_{B})\\}$
$\displaystyle\hskip 56.9055pt=\check{\bf W}_{\rm d}^{t}({\bf
k},t,t_{A})\\{\check{\bf A}_{\rm d}^{t}{\bf N}+{\bf N}\check{\bf A}_{\rm
d}\\}\check{\bf W}_{\rm d}({\bf k},t,t_{B})={\bf O},$ (108)

where ${\bf O}$ is the null-matrix. This expression confirms that $\check{\bf
W}_{\rm d}^{t}({\bf k},t,t_{A}){\bf N}\check{\bf W}_{\rm d}({\bf k},t,t_{B})$
is independent of $t$. This derivation holds for continuously varying medium
parameters. However, since the components of the propagator matrix are
continuous over a time-boundary, it follows that $\check{\bf W}_{\rm
d}^{t}({\bf k},t,t_{A}){\bf N}\check{\bf W}_{\rm d}({\bf k},t,t_{B})$ is a
propagation invariant for a time-dependent medium with piecewise continuous
parameters. Taking $t=t_{A}$ and subsequently $t=t_{B}$, we obtain

$\displaystyle\check{\bf W}_{\rm d}^{t}({\bf k},t_{A},t_{A}){\bf N}\check{\bf
W}_{\rm d}({\bf k},t_{A},t_{B})=\check{\bf W}_{\rm d}^{t}({\bf
k},t_{B},t_{A}){\bf N}\check{\bf W}_{\rm d}({\bf k},t_{B},t_{B}),$ (109)

or, using equation (104),

$\displaystyle{\bf N}\check{\bf W}_{\rm d}({\bf k},t_{A},t_{B})=\check{\bf
W}_{\rm d}^{t}({\bf k},t_{B},t_{A}){\bf N},$ (110)

or, applying the inverse 2D Fourier transformation defined in equation (37),

$\displaystyle{\bf N}{\bf W}_{\rm d}({\bf x},t_{A},t_{B})={\bf W}_{\rm
d}^{t}({\bf x},t_{B},t_{A}){\bf N}.$ (111)

Equations (110) and (111) formulate symmetry properties of the propagator
matrix in the ${\bf k},t$\- and ${\bf x},t$-domain, respectively. Substituting
equations (97) and (101) into equation (111) we find the following symmetry
properties for the components of the propagator matrix

$\displaystyle W_{\rm d}^{U,V}({\bf x},t_{A},t_{B})$ $\displaystyle=$
$\displaystyle-W_{\rm d}^{U,V}({\bf x},t_{B},t_{A}),$ (112) $\displaystyle
W_{\rm d}^{V,U}({\bf x},t_{A},t_{B})$ $\displaystyle=$ $\displaystyle-W_{\rm
d}^{V,U}({\bf x},t_{B},t_{A}),$ (113) $\displaystyle W_{\rm d}^{V,V}({\bf
x},t_{A},t_{B})$ $\displaystyle=$ $\displaystyle W_{\rm d}^{U,U}({\bf
x},t_{B},t_{A}).$ (114)

Next we show that three of the four components of the propagator matrix can be
expressed in terms of the upper-right component $W_{\rm d}^{U,V}({\bf
x},t,t_{0})$. From equations (93), (98) and (101) we find

$\displaystyle W_{\rm d}^{V,V}({\bf x},t,t_{0})$ $\displaystyle=$
$\displaystyle-\beta(t)\partial_{t}W_{\rm d}^{U,V}({\bf x},t,t_{0}),$ (115)
$\displaystyle W_{\rm d}^{V,U}({\bf x},t,t_{0})$ $\displaystyle=$
$\displaystyle-\beta(t)\partial_{t}W_{\rm d}^{U,U}({\bf x},t,t_{0}).$ (116)

From equations (114), (115) and (112) we find

$\displaystyle W_{\rm d}^{U,U}({\bf x},t,t_{0})$ $\displaystyle=$
$\displaystyle\beta(t_{0})\partial_{t_{0}}W_{\rm d}^{U,V}({\bf x},t,t_{0}).$
(117)

Substitution into equation (116) yields

$\displaystyle W_{\rm d}^{V,U}({\bf x},t,t_{0})$ $\displaystyle=$
$\displaystyle-\beta(t)\beta(t_{0})\partial_{t}\partial_{t_{0}}W_{\rm
d}^{U,V}({\bf x},t,t_{0}).$ (118)

Hence, equations (115), (117) and (118) are the sought relations.

###  11.4 Relations between the Green’s function and the propagator matrix

We show that the Green’s function ${\cal G}({\bf x},{\bf 0},t,t_{0})$, obeying
wave equation (30) for ${\bf x}_{0}={\bf 0}$ with causality condition (31),
can be expressed in terms of the upper-right component of the propagator
matrix, according to

$\displaystyle{\cal G}({\bf x},{\bf 0},t,t_{0})=-H(t-t_{0})W_{\rm
d}^{U,V}({\bf x},t,t_{0}).$ (119)

Note that this relation is different from that for the 1D situation (which
includes a spatial derivative on the left-hand side (Wapenaar et al., 2024))
because of the deviating definition of the wave field vector ${\bf q}_{\rm d}$
and consequently of the propagator matrix ${\bf W}_{\rm d}$, see sections 10
and 11.1.

Due to the Heaviside function in equation (119), the causality condition is
fulfilled. In the following we show that $H(t-t_{0})W_{\rm d}^{U,V}({\bf
x},t,t_{0})$ obeys the same wave equation as $-{\cal G}({\bf x},{\bf
0},t,t_{0})$. For the first time derivative we have

$\displaystyle\partial_{t}\\{H(t-t_{0})W_{\rm d}^{U,V}({\bf
x},t,t_{0})\\}=\delta(t-t_{0})W_{\rm d}^{U,V}({\bf
x},t,t_{0})+H(t-t_{0})\partial_{t}W_{\rm d}^{U,V}({\bf x},t,t_{0}).$ (120)

From equations (99) and (101) we have $W_{\rm d}^{U,V}({\bf
x},t_{0},t_{0})=0$. Using this and equation (115) in equation (120), we obtain

$\displaystyle\partial_{t}\\{H(t-t_{0})W_{\rm d}^{U,V}({\bf
x},t,t_{0})\\}=-\frac{1}{\beta(t)}H(t-t_{0})W_{\rm d}^{V,V}({\bf x},t,t_{0}).$
(121)

From equations (93), (98), (99) and (101) we have $\partial_{t}W_{\rm
d}^{V,V}({\bf x},t,t_{0})=-\frac{1}{\alpha(t)}{\bf\nabla}^{2}W_{\rm
d}^{U,V}({\bf x},t,t_{0})$ and $W_{\rm d}^{V,V}({\bf
x},t_{0},t_{0})=\delta({\bf x})$. Using this, we find for the second time
derivative

$\displaystyle\partial_{t}\bigl{(}\beta(t)\partial_{t}\\{H(t-t_{0})W_{\rm
d}^{U,V}({\bf x},t,t_{0})\\}\bigr{)}$ $\displaystyle=$
$\displaystyle-\partial_{t}\\{H(t-t_{0})W_{\rm d}^{V,V}({\bf x},t,t_{0})\\}$
$\displaystyle=$ $\displaystyle-\delta(t-t_{0})\delta({\bf
x})+\frac{1}{\alpha(t)}{\bf\nabla}^{2}\\{H(t-t_{0})W_{\rm d}^{U,V}({\bf
x},t,t_{0})\\}.$

Comparing this with equation (30) for ${\bf x}_{0}={\bf 0}$, with $c(t)$
defined in equation (10), we conclude that $H(t-t_{0})W_{\rm d}^{U,V}({\bf
x},t,t_{0})$ indeed obeys the same wave equation as $-{\cal G}({\bf x},{\bf
0},t,t_{0})$. This completes the proof of equation (119). Figure 4(b) shows a
$z,t$-diagram of the Green’s function ${\cal G}({\bf x},{\bf 0},t,t_{0})$
(convolved with a spatial wavelet) for $x=0$ and $t_{0}=25\,\mu$s. It has been
obtained from $W_{\rm d}^{U,V}({\bf x},t,t_{0})$ via equation (119). The green
line at $t_{0}=25\,\mu$s indicates the causality condition of equation (31).

In a similar way as above we can show that the acausal Green’s function ${\cal
G}^{a}({\bf x},{\bf 0},t,t_{0})$, with condition ${\cal G}^{a}({\bf x},{\bf
0},t,t_{0})=0$ for $t>t_{0}$, can be expressed as

$\displaystyle{\cal G}^{a}({\bf x},{\bf 0},t,t_{0})=H(t_{0}-t)W_{\rm
d}^{U,V}({\bf x},t,t_{0}).$ (123)

Vice versa, by combining equations (119) and (123), we can express $W_{\rm
d}^{U,V}({\bf x},t,t_{0})$ in terms of the Green’s function and its acausal
version, according to

$\displaystyle W_{\rm d}^{U,V}({\bf x},t,t_{0})={\cal G}^{a}({\bf x},{\bf
0},t,t_{0})-{\cal G}({\bf x},{\bf 0},t,t_{0}).$ (124)

Using equations (115), (117) and (118), the other components of the propagator
matrix ${\bf W}_{\rm d}({\bf x},t,t_{0})$ can also be expressed in terms of
the Green’s function and its acausal version.

##  12 Conclusions

We discussed fundamental aspects of wave propagation and scattering in 2D
homogeneous time-dependent materials, using a unified notation which
simultaneously captures electromagnetic, acoustic and elastodynamic shear
waves.

We reviewed transmission and reflection of a plane wave that is incident on a
time boundary. Due to causality, the transmitted and reflected waves both
exist only after the time boundary. The net field-momentum density is
conserved over a time boundary, but the net power-flux density is not.

We reviewed the Green’s function of a time-dependent material, being the
causal response to an impulsive point source. For the special case of a single
time boundary, the reflected part of the Green’s function after the time
boundary propagates back and focuses at the position of the original source.
Subsequently, the focus acts as a virtual source for a wave field with
opposite amplitude.

We discussed propagation invariants (i.e., time-independent quantities) for
homogeneous, piecewise continuous time-dependent materials. The net field-
momentum density is a special case of a propagation invariant: it is conserved
not only over time boundaries (as mentioned above), but also in the
continuously varying material between the time boundaries. More general
propagation invariants have been formulated for specific combinations of wave
fields in two mutually independent states. These propagation invariants form
the basis for general reciprocity theorems of the space-convolution type and
of the space-correlation type for homogeneous, piecewise continuous time-
dependent materials.

We used the reciprocity theorem of the space-convolution type to derive a
source-receiver reciprocity relation for the Green’s function of a time-
dependent material: the causal Green’s function between a source at $t_{B}$
and a receiver at $t_{A}$ is identical to the acausal Green’s function between
a sink at $t_{A}$ and a receiver at $t_{B}$. We also used the reciprocity
theorem of the space-convolution type to derive a general wave-field
representation for a homogeneous time-dependent material. Similar as its
counterpart for an inhomogeneous time-independent material, it forms the basis
for the analysis of wave scattering problems. As an example, we used this
representation to quantitatively explain the behaviour of the Green’s function
after a single time boundary. We used the reciprocity theorem of the space-
correlation type to derive a representation for Green’s function retrieval in
a homogeneous, piecewise continuous time-dependent material.

Finally, we formulated a matrix-vector wave equation for time-dependent
materials and used this as a basis for deriving a propagator matrix, its
symmetry properties and its relation with the causal and acausal Green’s
functions.

## Acknowledgements

The author thanks Johannes Aichele and Dirk-Jan van Manen for the inspiring
discussions about waves in time-dependent materials.

##  Appendix A Inverse Fourier Transform of wave-vector time domain Green’s
function

We evaluate the inverse Fourier transform of $\check{\cal G}({\bf k},{\bf
x}_{0},t,t_{0})$, defined in equation (45). Using equation (37), we write

$\displaystyle{\cal G}({\bf x},{\bf x}_{0},t,t_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\int_{{\mathbb{R}}^{2}}\check{\cal G}({\bf
k},{\bf x}_{0},t,t_{0})\exp(i{\bf k}\cdot{\bf x}){\rm d}{\bf k}$ (125)
$\displaystyle=$ $\displaystyle\frac{H(\Delta
t)}{4\pi^{2}\eta}\int_{{\mathbb{R}}^{2}}\frac{\sin(k_{r}c\Delta
t)}{k_{r}}\exp(i{\bf k}\cdot\Delta{\bf x}){\rm d}{\bf k},$

with $\Delta t=t-t_{0}$, $k_{r}=|{\bf k}|$ and $\Delta{\bf x}={\bf x}-{\bf
x}_{0}$. Defining polar coordinates ${\bf k}=k_{r}(\cos\theta,\sin\theta)$ and
$\Delta{\bf x}=r(\cos\phi,\sin\phi)$, with $r=|\Delta{\bf x}|$, we have ${\bf
k}\cdot\Delta{\bf x}=k_{r}r\cos(\theta-\phi)$ and ${\rm d}{\bf k}=k_{r}{\rm
d}k_{r}{\rm d}\theta$, hence

$\displaystyle{\cal G}({\bf x},{\bf x}_{0},t,t_{0})$ $\displaystyle=$
$\displaystyle\frac{H(\Delta t)}{2\pi\eta}\int_{0}^{\infty}{\rm
d}k_{r}\sin(k_{r}c\Delta
t)\frac{1}{2\pi}\int_{0}^{2\pi}\exp(ik_{r}r\cos(\theta-\phi)){\rm d}\theta.$
(126)

The integral over $\theta$ is independent of $\phi$, so we choose $\phi=0$.
Furthermore, this integral from $0$ to $2\pi$ is twice the real part of the
integral from $0$ to $\pi$. Hence

$\displaystyle{\cal G}({\bf x},{\bf x}_{0},t,t_{0})$ $\displaystyle=$
$\displaystyle\frac{H(\Delta t)}{2\pi\eta}\int_{0}^{\infty}{\rm
d}k_{r}\sin(k_{r}c\Delta
t)\frac{1}{\pi}\int_{0}^{\pi}\cos(k_{r}r\cos\theta){\rm d}\theta.$ (127)

Using the definition of the Bessel function $J_{0}$ in equation (9.1.18) of
(Abramowitz & Stegun, 1970), we obtain

$\displaystyle{\cal G}({\bf x},{\bf x}_{0},t,t_{0})$ $\displaystyle=$
$\displaystyle\frac{H(\Delta t)}{2\pi\eta}\int_{0}^{\infty}\sin(k_{r}c\Delta
t)J_{0}(k_{r}r){\rm d}k_{r}.$ (128)

Finally, using equation (6.671-7) of (Gradshteyn et al., 1994) and the
definitions of $\Delta t$ and $r$ given below equation (125) yields

$\displaystyle{\cal G}({\bf x},{\bf x}_{0},t,t_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\beta c^{2}}\frac{H\Bigl{(}t-t_{0}-\frac{|{\bf
x}-{\bf x}_{0}|}{c}\Bigr{)}}{\sqrt{(t-t_{0})^{2}-\frac{|{\bf x}-{\bf
x}_{0}|^{2}}{c^{2}}}}.$ (129)

##  Appendix B Propagator matrix for a time-independent slab

For the special case of a time-independent slab between $t_{n-1}$ and $t_{n}$,
with parameters $\alpha_{n}$ and $\beta_{n}$, the solution of equation (103),
with boundary condition (104) (with $t_{0}$ replaced by $t_{n-1}$ and $t$ by
$t_{n}$), reads

$\displaystyle\check{\bf W}_{\rm d}({\bf
k},t_{n},t_{n-1})=\begin{pmatrix}\cos(|{\bf k}|c_{n}\Delta
t_{n})&-\frac{1}{\eta_{n}|{\bf k}|}\sin(|{\bf k}|c_{n}\Delta t_{n})\\\
\eta_{n}|{\bf k}|\sin(|{\bf k}|c_{n}\Delta t_{n})&\cos(|{\bf k}|c_{n}\Delta
t_{n})\end{pmatrix},$ (130)

with $c_{n}=1/\sqrt{\alpha_{n}\beta_{n}}$, $\eta_{n}=\beta_{n}c_{n}$ and
$\Delta t_{n}=t_{n}-t_{n-1}$. Note that the upper-right component can,
according to the Fourier transform of equation (124), be expressed as

$\displaystyle\check{W}_{\rm d}^{U,V}({\bf k},t_{n},t_{n-1})=\check{\cal
G}^{a}({\bf k},{\bf 0},t_{n},t_{n-1})-\check{\cal G}({\bf k},{\bf
0},t_{n},t_{n-1}),$ (131)

with $\check{\cal G}({\bf k},{\bf 0},t_{n},t_{n-1})$ given in equation (45)
for ${\bf x}_{0}={\bf 0}$, $t_{0}=t_{n-1}$ and $t=t_{n}$. Similarly, we obtain
an explicit expression for $W_{\rm d}^{U,V}({\bf x},t_{n},t_{n-1})$ in the
space-time domain by substituting equation (129) for ${\bf x}_{0}={\bf 0}$
into equation (124). This gives

$\displaystyle W_{\rm d}^{U,V}({\bf x},t_{n},t_{n-1})$ $\displaystyle=$
$\displaystyle{\cal G}({\bf x},{\bf 0},t_{n-1},t_{n})-{\cal G}({\bf x},{\bf
0},t_{n},t_{n-1})$ (132) $\displaystyle=$
$\displaystyle\frac{1}{2\pi\beta_{n}c_{n}^{2}}\frac{H(-\Delta t_{n}-|{\bf
x}|/c_{n})-H(\Delta t_{n}-|{\bf x}|/c)}{\sqrt{\Delta t_{n}^{2}-|{\bf
x}|^{2}/c_{n}^{2}}}.$

Explicit expressions for the other components of ${\bf W}_{\rm d}({\bf
x},t_{n},t_{n-1})$ follow by substituting equation (132) into equations (115),
(117) and (118), with $t_{0}$ and $t$ replaced by $t_{n-1}$ and $t_{n}$,
respectively.

## References

  * Abramowitz & Stegun (1970) Abramowitz, M. & Stegun, I. A., 1970.  Handbook of mathematical functions, Dover Publications, Inc., New York.
  * Ammari et al. (2024) Ammari, H., Cao, J., Hiltunen, E. O., & Rueff, L., 2024.  Scattering from time-modulated subwavelength resonators, Proceedings of the Royal Society A, 480, 20240177.
  * Apffel & Fort (2022) Apffel, B. & Fort, E., 2022.  Frequency conversion cascade by crossing multiple space and time interfaces, Physical Review Letters, 128, 064501.
  * Bacot et al. (2016) Bacot, V., Labousse, M., Eddi, A., Fink, M., & Fort, E., 2016.  Time reversal and holography with spacetime transformations, Nature Physics, 12, 972–977.
  * Bal et al. (2019) Bal, G., Fink, M., & Pinaud, O., 2019.  Time-reversal by time-dependent perturbations, SIAM J. Appl. Math., 79, 754–780.
  * Berkhout (1985) Berkhout, A. J., 1985.  Seismic Migration. Imaging of acoustic energy by wave field extrapolation. A. Theoretical aspects (3rd edition), Elsevier, Amsterdam.
  * Bleistein (1984) Bleistein, N., 1984.  Mathematical methods for wave phenomena, Academic Press, Inc., Orlando.
  * Bojarski (1983) Bojarski, N. N., 1983.  Generalized reaction principles and reciprocity theorems for the wave equations and the relationship between the time-advanced and time-retarded fields, Journal of the Acoustical Society of America, 74, 281–285.
  * Burns et al. (2020) Burns, L., Bliokh, K. Y., Nori, F., & Dressel, J., 2020.  Acoustic versus electromagnetic field theory: scalar, vector, spinor representations and the emergence of acoustic spin, New Journal of Physics, 22, 053050.
  * Caloz & Deck-Léger (2020a) Caloz, C. & Deck-Léger, Z. L., 2020a.  Spacetime metamaterials—Part I: General concepts, IEEE Trans. Ant. Prop., 68(3), 1569–1582.
  * Caloz & Deck-Léger (2020b) Caloz, C. & Deck-Léger, Z. L., 2020b.  Spacetime metamaterials—Part II: Theory and applications, IEEE Trans. Ant. Prop., 68(3), 1583–1598.
  * Campillo & Paul (2003) Campillo, M. & Paul, A., 2003.  Long-range correlations in the diffuse seismic coda, Science, 299, 547–549.
  * Carcione & Cavallini (1995) Carcione, J. M. & Cavallini, F., 1995.  On the acoustic-electromagnetic analogy, Wave Motion, 21, 149–162.
  * Carcione & Robinson (2002) Carcione, J. M. & Robinson, E. A., 2002.  On the acoustic-electromagnetic analogy for the reflection-refraction problem, Studia Geophysica et Geodaetica, 46, 321–346.
  * Corones (1975) Corones, J. P., 1975.  Bremmer series that correct parabolic approximations, J. Math. Anal. Appl., 50, 361–372.
  * de Hoop (1958) de Hoop, A. T., 1958.  Representation theorems for the displacement in an elastic solid and their applications to elastodynamic diffraction theory, Ph.D. thesis, Delft University of Technology, Delft.
  * de Hoop (1966) de Hoop, A. T., 1966.  An elastodynamic reciprocity theorem for linear, viscoelastic media, Applied Scientific Research, 16, 39–45.
  * de Hoop (1995) de Hoop, A. T., 1995.  Handbook of radiation and scattering of waves, Academic Press, London.
  * de Hoop & Lager (2014) de Hoop, A. T. & Lager, I. E., 2014.  Closed-form analytic expressions for the pulsed-source radiated electromagnetic field in a class of media with time-varying wave speed, Wave Motion, 51, 418–424.
  * Druzhinin et al. (1998) Druzhinin, A., Pedersen, H., Campillo, M., & Kim, W., 1998.  Elastic Kirchhoff-Helmholtz synthetic seismograms, Pure and Applied Geophysics, 151, 17–45.
  * Feynman et al. (1963) Feynman, R., Leighton, R., & Sands, M., 1963.  The Feynman lectures on physics, Volume II, California Institute of Technology.
  * Fink & Fort (2017) Fink, M. & Fort, E., 2017.  From the time-reversal mirror to the instantaneous time mirror, Eur. Phys. J. Special Topics, 226, 1477–1486.
  * Fishman & McCoy (1984) Fishman, L. & McCoy, J. J., 1984.  Derivation and application of extended parabolic wave theories. I. The factorized Helmholtz equation, J. Math. Phys., 25(2), 285–296.
  * Fokkema & van den Berg (1993) Fokkema, J. T. & van den Berg, P. M., 1993.  Seismic applications of acoustic reciprocity, Elsevier, Amsterdam.
  * Frazer & Sen (1985) Frazer, L. N. & Sen, M. K., 1985.  Kirchhoff-Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium, I, Theory, Geophysical Journal of the Royal Astronomical Society, 80, 121–147.
  * Gangi (1970) Gangi, A. F., 1970.  A derivation of the seismic representation theorem using seismic reciprocity, Journal of Geophysical Research, 75, 2088–2095.
  * Garnier (2021) Garnier, J., 2021.  Wave propagation in periodic and random time-dependent media, SIAM Multiscale Model. Simul., 19, 1190–1211.
  * Gilbert & Backus (1966) Gilbert, F. & Backus, G. E., 1966.  Propagator matrices in elastic wave and vibration problems, Geophysics, 31(2), 326–332.
  * Goldsberry et al. (2019) Goldsberry, B. M., Wallen, S. P., & Haberman, M. R., 2019.  Non-reciprocal wave propagation in mechanically-modulated continuous elastic metamaterials, Journal of the Acoustical Society of America, 146(1), 782–788.
  * Gradshteyn et al. (1994) Gradshteyn, I. S., Ryzhik, I. M., & Jeffrey, A., 1994.  Table of integrals, series and products, Academic Press, Boston.
  * Haberman & Guild (2016) Haberman, M. R. & Guild, M. D., 2016.  Acoustic metamaterials, Physics Today, 69(6), 42–48.
  * Haines (1988) Haines, A. J., 1988.  Multi-source, multi-receiver synthetic seismograms for laterally heterogeneous media using F-K domain propagators, Geophysical Journal International, 95, 237–260.
  * Haney (2009) Haney, M. M., 2009.  Infrasonic ambient noise interferometry from correlations of microbaroms, Geophysical Research Letters, 36, L19808.
  * Haskell (1953) Haskell, N. A., 1953.  The dispersion of surface waves on multilayered media, Bulletin of the Seismological Society of America, 43, 17–34.
  * Hidalgo-Caballero et al. (2023) Hidalgo-Caballero, S., Sreenivas, S. K., Bacot, V., Wildeman, S., Harazi, M., Jia, X., Tourin, A., Fink, M., Cassinelli, A., Labousse, M., & Fort, E., 2023\.  Damping-driven time reversal for waves, Physical Review Letters, 130, 087201.
  * Innanen (2018) Innanen, K. A., 2018.  Potential monitoring applications of seismic reflections from time and mixed space/time boundaries, in SEG, Expanded Abstracts, pp. 3868–3872.
  * Jiang (1975) Jiang, C. L., 1975.  Wave propagation and dipole radiation in a suddenly created plasma, IEEE Trans. Ant. Prop., 23(1), 83–90.
  * Kennett (1972) Kennett, B. L. N., 1972.  Seismic waves in laterally inhomogeneous media, Geophysical Journal of the Royal Astronomical Society, 27, 301–325.
  * Kennett (1983) Kennett, B. L. N., 1983.  Seismic wave propagation in stratified media, Cambridge University Press.
  * Kennett et al. (1990) Kennett, B. L. N., Koketsu, K., & Haines, A. J., 1990.  Propagation invariants, reflection and transmission in anisotropic, laterally heterogeneous media, Geophysical Journal International, 103, 95–101.
  * Knopoff (1956) Knopoff, L., 1956.  Diffraction of elastic waves, Journal of the Acoustical Society of America, 28, 217–229.
  * Knopoff & Gangi (1959) Knopoff, L. & Gangi, A. F., 1959.  Seismic reciprocity, Geophysics, 24, 681–691.
  * Koketsu et al. (1991) Koketsu, K., Kennett, B. L. N., & Takenaka, H., 1991.  2-D reflectivity method and synthetic seismograms for irregularly layered structures - II. Invariant embedding approach, Geophysical Journal International, 105, 119–130.
  * Kosloff & Baysal (1983) Kosloff, D. D. & Baysal, E., 1983.  Migration with the full acoustic wave equation, Geophysics, 48(6), 677–687.
  * Koutserimpas & Fleury (2018) Koutserimpas, T. T. & Fleury, R., 2018.  Electromagnetic waves in a time periodic medium with step-varying refractive index, IEEE Trans. Ant. Prop., 66(10), 5300–5307.
  * Landau & Lifshitz (1960) Landau, L. D. & Lifshitz, E. M., 1960.  Electrodynamics of continuous media, Pergamon Press.
  * Lorentz (1895) Lorentz, H. A., 1895.  The theorem of Poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light, Verslagen der Afdeeling Natuurkunde van de Koninklijke Akademie van Wetenschappen, 4, 176–187.
  * Løseth & Ursin (2007) Løseth, L. O. & Ursin, B., 2007.  Electromagnetic fields in planarly layered anisotropic media, Geophysical Journal International, 170, 44–80.
  * Lurie (2007) Lurie, K. A., 2007.  An introduction to the mathematical theory of dynamic materials, Springer.
  * Malcolm et al. (2004) Malcolm, A. E., Scales, J. A., & van Tiggelen, B. A., 2004.  Extracting the Green function from diffuse, equipartitioned waves, Physical Review E, 70, 015601(R).
  * Mendonça & Shukla (2002) Mendonça, J. T. & Shukla, P. K., 2002.  Time refraction and time reflection: Two basic concepts, Physica Scripta, 65, 160–163.
  * Morgenthaler (1958) Morgenthaler, F. R., 1958.  Velocity modulation of electromagnetic waves, IRE Transactions on Microwave Theory and Techniques, 6(2), 167–172.
  * Morse & Feshbach (1953) Morse, P. M. & Feshbach, H., 1953.  Methods of theoretical physics, Vol. I, McGraw-Hill Book Company Inc., New York.
  * Mounaix et al. (2020) Mounaix, M., Fontaine, N. K., Neilson, D. T., Ryf, R., Chen, H., Alvarado-Zacarias, J. C., & Carpenter, J., 2020.  Time reversed optical waves by arbitrary vector spatiotemporal field generation, Nature Communications, 11, 5813.
  * Moussa et al. (2023) Moussa, H., Xu, G., Yin, S., Galiffi, E., Ra’di, Y., & Alù, A., 2023.  Observation of temporal reflection and broadband frequency translation at photonic time interfaces, Nature Physics, 19, 863–868.
  * Nassar et al. (2017a) Nassar, H., Chen, H., Norris, A. N., Haberman, M. R., & Huang, G. L., 2017a.  Non-reciprocal wave propagation in modulated elastic metamaterials, Proceedings of the Royal Society A, 473, 20170188.
  * Nassar et al. (2017b) Nassar, H., Xu, X. C., Norris, A. N., & Huang, G. L., 2017b.  Modulated phononic crystals: Non-reciprocal wave propagation and Willis materials, Journal of the Mechanics and Physics of Solids, 101, 10–29.
  * Pacheco-Peña & Engheta (2020) Pacheco-Peña, V. & Engheta, N., 2020.  Effective medium concept in temporal metamaterials, Nanophotonics, 9(2), 379–391.
  * Pao & Varatharajulu (1976) Pao, Y. H. & Varatharajulu, V., 1976.  Huygens’ principle, radiation conditions and integral formulations for the scattering of elastic waves, Journal of the Acoustical Society of America, 59, 1361–1371.
  * Peng et al. (2020) Peng, D., Fan, Y., Liu, R., Guo, X., & Wang, S., 2020.  Time-reversed water waves generated from an instantaneous time mirror, Journal of Physics Communications, 4(10), 105013.
  * Ptitcyn et al. (2023) Ptitcyn, G., Mirmoosa, M. S., Sotoodehfar, A., & Tretyakov, S. A., 2023.  A tutorial on the basics of time-varying electromagnetic systems and circuits: Historic overview and basic concepts of time-modulation, IEEE Antennas and Propagation Magazine, 65(4), 10–20.
  * Ramaccia et al. (2020) Ramaccia, D., Toscano, A., & Bilotti, F., 2020.  Light propagation through metamaterial temporal slabs: reflection, refraction, and special cases, Optics Letters, 45(20), 5836–5839.
  * Rayleigh (1878) Rayleigh, J. W. S., 1878.  The theory of sound. Volume II, Dover Publications, Inc. (Reprint 1945).
  * Roux et al. (2004) Roux, P., Kuperman, W. A., & the NPAL Group, 2004.  Extracting coherent wave fronts from acoustic ambient noise in the ocean, Journal of the Acoustical Society of America, 116, 1995–2003.
  * Sabra et al. (2007) Sabra, K. G., Conti, S., Roux, P., & Kuperman, W. A., 2007.  Passive in vivo elastography from skeletal muscle noise, Applied Physics Letters, 90, 194101.
  * Salem & Caloz (2015) Salem, M. A. & Caloz, C., 2015.  Space-time cross-mapping and application to wave scattering, doi.org/10.48550/arXiv.1504.02012.
  * Snieder (2004) Snieder, R., 2004.  Extracting the Green’s function from the correlation of coda waves: A derivation based on stationary phase, Physical Review E, 69, 046610.
  * Sotoodehfar et al. (2023) Sotoodehfar, A., Boshgazi, S., Memarian, M., Rejaei, B., & Mehrany, K., 2023.  Investigating non-reciprocity in time-periodic media using a perturbative approach, Optics Express, 31(8), 12534–12548.
  * Takenaka et al. (1993) Takenaka, H., Kennett, B. L. N., & Koketsu, K., 1993.  The integral operator representation of propagation invariants for elastic waves in irregularly layered media, Wave Motion, 17, 299–317.
  * Thomson (1950) Thomson, W. T., 1950.  Transmission of elastic waves through a stratified solid medium, Journal of Applied Physics, 21, 89–93.
  * Torrent et al. (2018) Torrent, D., Parnell, W. J., & Norris, A. N., 2018.  Loss compensation in time-dependent elastic metamaterials, Phys. Rev. B, 97, 014105.
  * Trainiti & Ruzzene (2016) Trainiti, G. & Ruzzene, M., 2016.  Non-reciprocal elastic wave propagation in spatiotemporal periodic structures, New Journal of Physics, 18, 083047.
  * Tygel et al. (2000) Tygel, M., Schleicher, J., Santos, L. T., & Hubral, P., 2000.  An asymptotic inverse to the Kirchhoff-Helmholtz integral, Inverse Problems, 16, 425–445.
  * Ursin (1983) Ursin, B., 1983.  Review of elastic and electromagnetic wave propagation in horizontally layered media, Geophysics, 48, 1063–1081.
  * Van Manen et al. (2024) Van Manen, D. J., Aichele, J., Müller, J., Serra-Garcia, M., & Wapenaar, K., 2024\.  On acoustic space-time media that compute their own inverse, doi.org/10.48550/arXiv.2406.15203.
  * Wapenaar & Berkhout (1986) Wapenaar, C. P. A. & Berkhout, A. J., 1986.  Wave-field extrapolation techniques for inhomogeneous media which include critical angle events. Part II: Methods using the two-way wave equation, Geophysical Prospecting, 34(2), 147–179.
  * Wapenaar et al. (2001) Wapenaar, C. P. A., Dillen, M. W. P., & Fokkema, J. T., 2001.  Reciprocity theorems for electromagnetic or acoustic one-way wave fields in dissipative inhomogeneous media, Radio Science, 36, 851–863.
  * Wapenaar (2003) Wapenaar, K., 2003.  Synthesis of an inhomogeneous medium from its acoustic transmission response, Geophysics, 68, 1756–1759.
  * Wapenaar & Fokkema (2006) Wapenaar, K. & Fokkema, J., 2006.  Green’s function representations for seismic interferometry, Geophysics, 71(4), SI33–SI46.
  * Wapenaar et al. (2004) Wapenaar, K., Thorbecke, J., & Draganov, D., 2004.  Relations between reflection and transmission responses of three-dimensional inhomogeneous media, Geophysical Journal International, 156, 179–194.
  * Wapenaar et al. (2024) Wapenaar, K., Aichele, J., & van Manen, D. J., 2024.  Waves in space-dependent and time-dependent materials: a systematic comparison, doi.org/10.48550/arXiv.2311.13428 (to appear in Wave Motion).
  * Weaver & Lobkis (2001) Weaver, R. L. & Lobkis, O. I., 2001.  Ultrasonics without a source: Thermal fluctuation correlations at MHz frequencies, Physical Review Letters, 87, 134301.
  * Willis (2011) Willis, J. R., 2011.  Effective constitutive relations for waves in composites and metamaterials, Proceedings of the Royal Society A, 467, 1865–1879.
  * Willis (2012) Willis, J. R., 2012.  The construction of effective relations for waves in a composite, Comptes Rendus Mecanique, 340, 181–192.
  * Woodhouse (1974) Woodhouse, J. H., 1974.  Surface waves in a laterally varying layered structure, Geophysical Journal of the Royal Astronomical Society, 37, 461–490.
  * Xiao et al. (2014) Xiao, Y., Maywar, D. N., & Agrawal, G. P., 2014.  Reflection and transmission of electromagnetic waves at a temporal boundary, Optics Letters, 39(3), 574–577.