# Edge Waves and Transmissions for Temporal
Laminates and Imperfect Chiral Interfaces

A.B. Movchan1, N.V. Movchan1, I.S. Jones1,2,
G.W. Milton3, H.-M. Nguyen4

1 Department of Mathematical Sciences, University of Liverpool, Liverpool L69
7ZL, U.K.
2 Liverpool John Moores University, Liverpool L3 3AF, U.K.
3 Department of Mathematics, University of Utah, Salt Lake City, Utah 84112,
U.S.A.
4 Laboratoire Jacques Louis Lions, Sorbonne Université, 4 Place Jussieu,
75013, Paris, France

###### Abstract

The analysis of wave patterns in a structure which possesses periodicity in
the spatial and temporal dimensions is presented. The topic of imperfect
chiral interfaces is also considered. Although causality is fundamental for
physical processes, natural wave phenomena can be observed when a wave is
split at a temporal interface. A wave split at a spatial interface is a more
common occurrence; however when the coefficients of the governing equations
are time-dependent, the temporal interface becomes important. Here, the
associated edge waves are studied, and regimes are analysed where the growth
of the solution in time is found. Imperfect interfaces, across which the
displacements are discontinuous, are also considered in the vector case of
chiral elastic systems. Analytical study and asymptotic approximations are
supplied with illustrative numerical examples.

##  1 Introduction

The phenomenon of wave reflection on spatial interfaces, separating two media,
is well-known and is described in classical textbooks (see, for example, books
by Kittel [1] and Lekner [2]). If the interfaces are arranged in a finite or
infinite array, then multiple reflections may lead to interesting phenomena of
dispersion connected to Floquet theory (see [1]). For finite stacks, the book
[2] has provided constructive algorithms for evaluation of transmission and
reflection characteristics of the structured stack.

The emphasis of the present paper is on interfaces of a different kind -
temporal interfaces. Interest in this area has been growing for a number of
years: see, for example, the book by Lurie [3] and the review of Caloz $\&$
Deck-Léger [4]. Most notable has been the work of Fink [5], and Bacot et al.
[6] on image recovery through partial time reversal at temporal interfaces.
The recent development in modelling of waves in media with temporal and
spatial interfaces was stimulated by the work of Lurie [3], Milton $\&$ Mattei
[7], and Mattei $\&$ Milton [8, 9], who introduced the key ideas in this novel
subject area.

With causality in place, a temporal interface still produces wave splitting so
that, at a certain time when the elastic or inertial properties of the carrier
medium change instantaneously, the wave front will split into two fronts,
propagating in different directions. Special combinations of temporal and
spatial interfaces can be analysed, where highly non-trivial wave patterns are
observed, as discussed in [7, 8, 9]. These patterns of characteristic lines
are known as “field patterns”.

These field patterns can arise in space-time geometries as simple as certain
temporal laminates. Here we show that this allows a straightforward
computation of their response which, in particular, reveals the behaviour at
the edge of the wave. Additionally, the case of chiral interfaces is explored.
The idea of non-reciprocity for dynamic materials was discussed by Lurie [3]
for the case of a laminate with oblique interfaces in space-time, and by Brun
et al. [10] in the context of chiral elastic lattices and homogenised coupled
elastic systems, This was followed by a computational experiment by Wang et
al. [11] who observed a uni-directional edge wave in a homogenised elastic
solid. The work on non-reciprocal dynamic materials was further stimulated by
novel physical applications and, in particular, by the concept of “topological
insulators”, as discussed by Moore [12], Pendry et al. [13], Hibbins et al.
[14], and Zhao et al. [15].

Floquet theory for transient problems in media with periodic temporal
interfaces can be successfully applied, as discussed by Nassar et al. [16],
and by Lurie $\&$ Weekes [17].

Recent work on modelling of chiral waves in elastic lattices by Carta et al.
[18, 19], has provided an explanation of the dynamics of discrete gyroscopic
systems in the context of dispersion, localisation and dynamic degeneracies.
The paper by Nieves et al. [20] provided mathematical insight into the
vibrations of chiral multi-structures and connections between the discrete
non-reciprocal systems and their continuous counterparts. The recent paper by
Jones et al. [21] has presented a comprehensive study of coupled elastic
waves, dynamic localisation, and dynamic Green’s functions in a chiral elastic
system. Furthermore, this work laid the foundations for the idea of an
imperfect temporal interface associated with high-gyricity material. One
characteristic feature of a gyroscopic force is that it is orthogonal to the
velocity vector, and hence a rotation transformation is involved in the
description of vector chiral waves and also in the analysis of imperfect
temporal chiral interfaces.

The structure of the paper is as follows: In Section 2 the formal description
of temporal interfaces and temporally stratified media is introduced. For a
given set of initial conditions, Section 3.1 presents the description of the
field patterns, associated with temporal laminates, and initiates the study of
edge waves. Special attention is given to the temporal switching algorithm,
which leads to the edge wave blow-up. For a different physical configuration
related to temporal interfaces, the edge waves were also observed in [17]. The
case of periodic initial conditions is considered in Section 4. The notion of
chiral vibrations and chiral interfaces is introduced in Section 5 and
includes the treatment of both spatial and temporal interfaces. Finally, the
derivation of the transmission conditions for chiral interfaces, characterised
by high gyricity, is presented in Section 6. Furthermore, the discussion of
the wave splitting and the edge wave is also presented. In that section, the
mathematical formulation is set as a vector problem and chiral temporal
interfaces provide coupling between the longitudinal and transverse
displacements of the wave field. Illustrative examples, which include closed
form solutions of the Cauchy problems in temporally stratified media, are
discussed in Section 7.

##  2 Formal settings - temporal interfaces

Let two-dimensional vector fields ${\bf j}$ and ${\bf e}$ be divergence-free
and curl-free, respectively in ${\bf R}^{2}$. Assume that

${\bf j}({\bf x})=\mbox{\boldmath${\sigma}$}({\bf x}){\bf e}({\bf x}),$ (1)

where

$\nabla\cdot{\bf j}({\bf x})=0,\leavevmode\nobreak\ {\bf e}=-\nabla V,$ (2)

with the diagonal matrix $\mbox{\boldmath${\sigma}$}({\bf
x})=\mbox{diag}(\alpha({\bf x}),-\beta({\bf x})).$ Here, $\alpha$ and $\beta$
are positive functions.

Thus, the potential function $V$ satisfies the governing equation

$\frac{\partial}{\partial x_{1}}\Bigg{(}\alpha({\bf
x})\frac{\partial}{\partial x_{1}}V({\bf x})\Bigg{)}-\frac{\partial}{\partial
x_{2}}\Bigg{(}\beta({\bf x})\frac{\partial}{\partial x_{2}}V({\bf
x})\Bigg{)}=0.$ (3)

The initial conditions are set at the boundary of the upper half-plane:

$V(x_{1},0)=\Phi(x_{1}),\leavevmode\nobreak\ \frac{\partial V}{\partial
x_{2}}(x_{1},0)=\Psi(x_{1}),\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{as}\leavevmode\nobreak\ x_{1}\in(-\infty,+\infty),$ (4)

where $\Phi$ and $\Psi$ are given bounded functions.

In particular, choose the independent variables $x_{1}$ and $x_{2}$ to be a
length coordinate $X$ and a time variable $T$, respectively, and refer to the
above problem (3)–(4) as the Cauchy problem for the wave equation in a
temporally and spatially inhomogeneous elastic string. In this case, $V$
stands for the elastic displacement, and the relations (4) represent the
initial conditions for the displacement and velocity at $T=0.$

###  2.1 Temporally inhomogeneous medium

Figure 1: Temporal laminate

Consider the special case where $\alpha$ and $\beta$ are $X-$independent, but
may change with $T$, i.e. $\alpha=\alpha(T),\leavevmode\nobreak\
\beta=\beta(T)$. In the context of the physical interpretation, related to
vibration of an elastic string, which extends along the $X-$axis, the
coefficients $\alpha$ and $\beta$ represent the elastic stiffness and the mass
density, respectively. Firstly, consider the case when these are continuous
and bounded functions of $T$. The wave equation (3) then becomes

$\alpha(T)\frac{\partial^{2}}{\partial X^{2}}V(X,T)-\frac{\partial}{\partial
T}\Bigg{(}\beta(T)\frac{\partial}{\partial T}V(X,T)\Bigg{)}=0.$ (5)

It is assumed that the materials are non-dispersive, since otherwise the
governing equation (5) would take a different form (see, for example, [22]).
Note that in (5) by interchanging $T$ and $X$ we obtain the governing equation
in a spatial laminate. This allows one to map results for spatial laminates to
temporal laminates. The difference is that one has to respect causality: an
instantaneous point source can only generate waves in the future, not the
past. The response to such point sources, or periodic arrays of them, will be
the main focus of our investigations in subsequent sections.

If one has a band gap in the spatial laminate then, in the temporal laminate,
one may have, for a fixed wavevector and for a frequency in the bandgap,
complex conjugate solutions, one of which corresponds to waves growing
exponentially in time.

The formal application of the Fourier transform

$\hat{V}(T,k)=\int_{-\infty}^{\infty}V(X,T)e^{iXk}dX$ (6)

leads to

$\alpha(T)k^{2}\hat{V}(k,T)+\frac{\partial}{\partial
T}\Bigg{(}\beta(T)\frac{d}{dT}\hat{V}(k,T)\Bigg{)}=0.$ (7)

In particular, if $\beta=1$, and the normalised stiffness coefficient $\alpha$
is a periodic function of time, for example
$\alpha(T)=1-2q\cos(2T),\leavevmode\nobreak\ 0<q<1/2$, then the equation (7)
becomes

$\frac{d^{2}}{dT^{2}}\hat{V}(k,T)+k^{2}(1-2q\cos(2T))\hat{V}(k,T)=0,$

which is the well-known Mathieu differential equation. The solution can be
represented in terms of the Mathieu functions, and Floquet theory can be used
accordingly (see, for example, [24]).

Consider an alternative configuration, where both the stiffness coefficient
$\alpha$ and the mass density coefficient $\beta$ are piece-wise constant
functions of $T$, which implies a “temporal stratification” (see Fig. 1). In
this case, the interface conditions should be set at every temporal interface,
as discussed in Section 2.2.

In the general case, equation (7) can be reduced to

$\frac{\partial}{\partial
T}\begin{pmatrix}\hat{W}\cr\hat{V}\end{pmatrix}+\begin{pmatrix}0&-\tilde{\alpha}(k,T)\cr({\beta}(T))^{-1}&0\end{pmatrix}\begin{pmatrix}\hat{W}\cr\hat{V}\end{pmatrix}=0,$
(8)

where

$\tilde{\alpha}(\xi,T)=k^{2}\alpha(T)\leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\ \hat{W}(k,T)=-\beta(T)\frac{\partial}{\partial
T}\hat{V}(k,T).$

The relations (4) yield the initial conditions for $\hat{W}$ and $\hat{V}$:

$\hat{V}\Bigg{|}_{T=0}=\hat{\Phi}(k),\leavevmode\nobreak\
\hat{W}\Bigg{|}_{T=0}=\beta(0)\hat{\Psi}(k),$ (9)

where $\hat{\Phi}(k),\hat{\Psi}(k)$ are the Fourier transforms of the right-
hand sides in (4).

Using the matrix notations

${\bf Y}(T)=\begin{pmatrix}\hat{W}\cr\hat{V}\end{pmatrix},\leavevmode\nobreak\
{\mbox{\boldmath${\cal
M}$}}(T)=\begin{pmatrix}0&-\tilde{\alpha}(k,T)\cr({\beta}(T))^{-1}&0\end{pmatrix},$
(10)

and regarding $k$ as a fixed parameter, we have the following initial boundary
value problem

$\frac{d}{dT}{\bf Y}(T)=-{\mbox{\boldmath${\cal M}$}}(T){\bf Y}(T),$ (11)
${\bf Y}(0)={\bf
Y}_{0}=\begin{pmatrix}\hat{W}\cr\hat{V}\end{pmatrix}\Bigg{|}_{T=0}.$ (12)

Hence, the vector function ${\bf Y}(T)$ evaluated at $T=T_{*}$ can be written
in the form

${\bf Y}(T_{*})=\exp(-\int_{0}^{T_{*}}{\mbox{\boldmath${\cal
M}$}}(\tau)d\tau){\bf Y}_{0}.$ (13)

In the case where ${\mbox{\boldmath${\cal M}$}}(\tau)$ is piecewise constant
in $\tau$ the exponential term can be written as a product of matrices. This
corresponds to the transfer matrix approach in spatial laminates.

###  2.2 Temporal stratification

Consider a horizontally stratified half-plane ${\bf
R}^{2}_{+}=\\{(X,T):X\in{\bf R},\leavevmode\nobreak\
T>0\\}=\overline{\cup_{j=1}^{\infty}\Omega_{j}}$, as shown in Fig. 1, where
$\Omega_{j}$ are the horizontal non-intersecting layers. Assume that across
the horizontal interfaces separating $\Omega_{n}$ and $\Omega_{n+1}$ the
functions $V$ and $W$ are continuous.

Let us introduce the set of positive constants $T_{1},T_{2}$ in such a way
that the straight line $\\{(X,T):X\in{\bf R},T=T_{1}\\}$ represents the
interface separating the adjacent strips $\Omega_{1}$ and $\Omega_{2}$, and
the straight line $\\{(X,T):X\in{\bf R},T=T_{1}+T_{2}\\}$ represents the
interface separating the adjacent strips $\Omega_{2}$ and $\Omega_{3}$. The
stratified structure is assumed to be periodic so that the line
$\\{(X,T):X\in{\bf R},T=n(T_{1}+T_{2})\\}$ separates the strips $\Omega_{2n}$
and $\Omega_{2n+1}$, and the line $\\{(X,T):X\in{\bf
R},T=nT_{1}+(n-1)T_{2})\\}$ separates the strips $\Omega_{2n-1}$ and
$\Omega_{2n}$. Also assume that the functions $\alpha(T)$ and $\beta(T)$ are
piecewise constant, i.e. $\alpha=\alpha_{1}$ and $\beta=\beta_{1}$ when
$(X,T)\in\Omega_{2n-1}$, and $\alpha=\alpha_{2}$ and $\beta=\beta_{2}$ when
$(X,T)\in\Omega_{2n}$ for all $n\in{\bf N}$.

To obtain a field pattern, and simplify the analysis, we consider the case
where

$T_{1}\sqrt{\alpha_{1}/\beta_{1}}=T_{2}\sqrt{\alpha_{2}/\beta_{2}}=d.$ (14)

This implies that the wave propagating with the speed of
$\sqrt{\alpha_{1}/\beta_{1}}$ covers during the time $T_{1}$ the same distance
$d$ as does the wave, which propagates with the speed
$\sqrt{\alpha_{2}/\beta_{2}}$ during the time $T_{2}$.

In this case, for a strip $\Omega_{n}$ equation (13) implies

${\bf
Y}(T^{(n)}_{+})=\frac{1}{2}\begin{pmatrix}e^{id|\xi|}+e^{-id|\xi|}&-i|\xi|\sqrt{\alpha\beta}(e^{id|\xi|}-e^{-id|\xi|})\cr\frac{i}{|\xi|\sqrt{\alpha\beta}}(e^{id|\xi|}-e^{-id|\xi|})&e^{id|\xi|}+e^{-id|\xi|}\end{pmatrix}{\bf
Y}(T^{(n)}_{-}),$ (15)

where $\alpha\beta$ should be replaced by $\alpha_{1}\beta_{1}$ for odd
$\Omega_{n}$ and by $\alpha_{2}\beta_{2}$ for even $\Omega_{n}$, respectively;
the values $T^{(n)}_{-}$ and $T^{(n)}_{+}$ stand for the lower and upper
limits of $T$ corresponding to the boundaries of the strip $\Omega_{n}$. The
matrix in the right-hand side of (15) has unit determinant.

This gives an iterative scheme enabling the Fourier transform of the solution
to be found in the temporally stratified half-plane, with wave reflections at
the temporal interfaces.

Figure 2: Wave splitting at temporal interfaces (dashed lines) showing the
emergent field pattern and the two edge waves (bold lines). The parameter
values are $\alpha_{1}/\beta_{1}=1,\alpha_{2}/\beta_{2}=4,T_{1}=1,T_{2}=1/2.$

##  3 Field patterns and the edge wave

Consider an example demonstrating a special feature of the wave, corresponding
to the periodic temporal pattern described above. For the sake of simplicity,
assume that $\Psi\equiv 0$ in (4). In this case, when $(X,T)\in\Omega_{1}$
(i.e. $0<T<T_{1}$) the field $V$ is given by

$V(X,T)=\frac{1}{2}\Bigg{(}\Phi(X-\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T)+\Phi(X+\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T)\Bigg{)},\leavevmode\nobreak\
\leavevmode\nobreak\ 0<T<T_{1},$ (16)

###  3.1 The edge wave

The transmission conditions at $T=T_{1}$ are continuity of the field $V$ and
”linear momentum” ($\beta\frac{\partial V}{\partial t}$). Application of these
conditions leads to the following form of the field $V$ in $\Omega_{2}$, where
$T_{1}<T<T_{1}+T_{2}$

$\displaystyle
V(X,T)=\frac{1}{4}\Bigg{\\{}(1+\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}})\Bigg{(}\Phi(X-\sqrt{\frac{\alpha_{2}}{\beta_{2}}}\tau-d)+\Phi(X+\sqrt{\frac{\alpha_{2}}{\beta_{2}}}\tau+d)\Bigg{)}$
$\displaystyle+(1-\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}})\Bigg{(}\Phi(X-\sqrt{\frac{\alpha_{2}}{\beta_{2}}}\tau+d)+\Phi(X+\sqrt{\frac{\alpha_{2}}{\beta_{2}}}\tau-d)\Bigg{)}\Bigg{\\}},$
(17)

where the quantity $d$ is defined in (14) and $\tau=T-T_{1}$.

The straightforward observation is that at the temporal interface the wave
$V(X,T)$ splits into the “edge wave”

$\frac{1}{4}(1+\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}})\Bigg{(}\Phi(X-\sqrt{\frac{\alpha_{2}}{\beta_{2}}}\tau-d)+\Phi(X+\sqrt{\frac{\alpha_{2}}{\beta_{2}}}\tau+d)\Bigg{)}$

and the “backward wave”

$\frac{1}{4}(1-\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}})\Bigg{(}\Phi(X-\sqrt{\frac{\alpha_{2}}{\beta_{2}}}\tau+d)+\Phi(X+\sqrt{\frac{\alpha_{2}}{\beta_{2}}}\tau-d)\Bigg{)},$

with the coefficient of the “edge wave” being greater than the coefficient of
the “backward wave” within the strip $\Omega_{2}$. We also note that the
“backward wave” is absent when the coefficients $\alpha,\beta$ satisfy the
relation $\alpha_{1}\beta_{1}=\alpha_{2}\beta_{2}$ corresponding to “matched
impedances”.

(a) (b)

Figure 3: The solution of the Cauchy problem for the case of temporal
stratification: The following values of the parameters were used in the
computations:
$\alpha_{1}=8,\alpha_{2}=3,\beta_{1}=0.7,\beta_{2}=0.5906,\kappa=3.1605,T_{1}=2,T_{2}=3.$
The initial profile is given by $\Phi(x)=e^{-x^{2}/10}$. (a) The displacement
$V(x,t)$, which takes into account both the spatial and temporal dependence.
(b) The oscillatory displacement $V$, corresponding to the edge wave; note the
exponential blow-up as $t$ increases: the dots show the representative values
of the displacement in the odd and even laminates, whereas the continuous
curve shows the oscillatory behaviour along the edge wave, also shown in the
part (a).

###  3.2 Further splitting at the temporal interface

The temporal interface splitting is illustrated in Fig. 2, where ray-branching
occurs at the times when $\alpha$ and $\beta$ switch values between
$\alpha_{1}$ and $\alpha_{2}$, and $\beta_{1}$ and $\beta_{2}$, respectively.

The next temporal interface is at $T=T_{1}+T_{2}.$ By application of the
transmission conditions of continuity of the field and ”linear momentum” at
the temporal interface $T=T_{1}+T_{2}$, the field $V(X,T)$ in the strip
$\Omega_{3}$, when $T_{1}+T_{2}<T<2T_{1}+T_{2}$, is given by

$\displaystyle
V(X,T)=\frac{1}{8}\Bigg{\\{}(\sqrt{\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}}-\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}})\Bigg{[}\Phi\Bigg{(}X+\sqrt{\frac{\alpha_{1}}{\beta_{1}}}(T-T_{1}-T_{2})\Bigg{)}$
$\displaystyle+\Phi\Bigg{(}X-\sqrt{\frac{\alpha_{1}}{\beta_{1}}}(T-T_{1}-T_{2})\Bigg{)}\Bigg{]}$
$\displaystyle+(2+\sqrt{\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}}+\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}})\Bigg{[}\Phi\Bigg{(}X+\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T+(\sqrt{\frac{\alpha_{2}}{\beta_{2}}}-\sqrt{\frac{\alpha_{1}}{\beta_{1}}})T_{2}\Bigg{)}$
$\displaystyle+\Phi\Bigg{(}X-\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T-(\sqrt{\frac{\alpha_{2}}{\beta_{2}}}-\sqrt{\frac{\alpha_{1}}{\beta_{1}}})T_{2})\Bigg{)}\Bigg{]}$
$\displaystyle+(2-\sqrt{\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}}-\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}})\Bigg{[}\Phi\Bigg{(}X+\sqrt{\frac{\alpha_{1}}{\beta_{1}}}(T-T_{1}-T_{2})\Bigg{)}$
$\displaystyle+\Phi\Bigg{(}X-\sqrt{\frac{\alpha_{1}}{\beta_{1}}}(T-T_{1}-T_{2})\Bigg{)}\Bigg{]}$
$\displaystyle-(\sqrt{\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}}-\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}})\Bigg{[}\Phi\Bigg{(}X-\sqrt{\frac{\alpha_{1}}{\beta_{1}}}(T-T_{1}-T_{2})+2H\Bigg{)}$
$\displaystyle+\Phi\Bigg{(}X+\sqrt{\frac{\alpha_{1}}{\beta_{1}}}(T-T_{1}-T_{2})-2H\Bigg{)}\Bigg{]}\Bigg{\\}}.$
(18)

We note that the “edge wave” term (in the above formula)

$\frac{1}{8}(2+\sqrt{\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}}+\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}})\Bigg{[}\Phi\Bigg{(}X+\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T+(\sqrt{\frac{\alpha_{2}}{\beta_{2}}}-\sqrt{\frac{\alpha_{1}}{\beta_{1}}})T_{2}\Bigg{)}$
$+\Phi\Bigg{(}X-\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T-(\sqrt{\frac{\alpha_{2}}{\beta_{2}}}-\sqrt{\frac{\alpha_{1}}{\beta_{1}}})T_{2})\Bigg{)}\Bigg{]}$

has the largest coefficient

$\frac{1}{8}(2+\sqrt{\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}}+\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}})$

compared to other coefficients in the representation resulting from the
temporal interface split; we also note that some of the coefficients in (18)
are negative.

###  3.3 The edge wave blow-up

The process may be applied to further layers to give a more detailed solution
of the Cauchy problem at any given time. The results are shown in Fig. 3a,
where the “edge wave” is clearly visible. Although the magnitude of the “edge
wave” is not monotonic in time, it may be seen that after passing $n$ macro-
cells $\Omega_{1}^{(j)}\cap\Omega_{2}^{(j)},\leavevmode\nobreak\
j=1,\ldots,n$, we have the following amplitude coefficients

${\cal
C}^{(1)}_{n}=\frac{1}{2}\Bigg{(}1+\frac{1}{4}(\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}}+\sqrt{\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}}-2)\Bigg{)}^{n}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{in}\leavevmode\nobreak\ \leavevmode\nobreak\ \Omega_{1}^{(n+1)},$ (19)
${\cal
C}^{(2)}_{n}=\frac{1}{2}\Big{(}1+\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}}\Big{)}{\cal
C}^{(1)}_{n}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{in}\leavevmode\nobreak\ \leavevmode\nobreak\
\Omega_{2}^{(n+1)},$ (20)

which both grow exponentially, as $n\to\infty$ for all cases where the
positive coefficients $\alpha$ and $\beta$ are chosen in such a way that
$\alpha_{1}\beta_{1}\neq\alpha_{2}\beta_{2}$. The formulae (19) and (20),
characterising the exponential growth of the edge wave amplitude in time, are
illustrated in Fig. 3b.

##  4 Field patterns for periodic initial conditions

Another example involving periodic initial conditions, leading to possible
growth of the solution due to wave splitting at temporal interfaces, is
considered below.

Consider initial conditions (4), with $\Phi=0$, while

$\Psi(X)=\sum_{k=-\infty}^{\infty}(-1)^{k}\delta(X-kD),$

with $D$ being a positive parameter, which represents half a period of the
field pattern along the $X-$axis. Here $\delta(X)$ is the Dirac delta
function.

###  4.1 “Reflection” at temporal interfaces

For the time interval $(0,T_{1})$, it is assumed that $\alpha=\alpha_{1}$,
$\beta=\beta_{1}$ are constant. Thus, when $0<T<T_{1}$ the field $V$ takes the
form

$V(X,T)=\frac{1}{2}\sqrt{\frac{\beta_{1}}{\alpha_{1}}}\sum_{k=-\infty}^{\infty}(-1)^{k}\Bigg{\\{}H(X-kD+\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T)-H(X-kD-\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T)\Bigg{\\}}.$
(21)

The derivation is similar to the one discussed in Section 3. When
$T_{1}<T<T_{1}+T_{2}$ we assume that $\alpha=\alpha_{2}$, $\beta=\beta_{2}$
take new constant values, so that

$T_{2}\sqrt{\alpha_{2}/\beta_{2}}=T_{1}\sqrt{\alpha_{1}/\beta_{1}}.$

Let $\tau=T-T_{1}$ and consider the field $V(X,T)$ in $\Omega_{2}$, where
$T_{1}<T<T_{1}+T_{2}$ in the form

$V(X,T)=W_{I}(X-\sqrt{\frac{\alpha_{2}}{\beta_{2}}}\tau)+W_{II}(X+\sqrt{\frac{\alpha_{2}}{\beta_{2}}}\tau),$
(22)

where the functions $W_{I}$ and $W_{II}$ represent the waves propagating in
the positive and negative direction of the $X-$axis. The transmission
conditions, representing the continuity of the “displacement” $V$ and of the
“linear momentum”, are set at the temporal interface
$T=T_{1}\leavevmode\nobreak\ (\tau=0)$:

$W_{I}(X)+W_{II}(X)=\frac{1}{2}\sqrt{\frac{\beta_{1}}{\alpha_{1}}}\sum_{k=-\infty}^{\infty}(-1)^{k}\Bigg{\\{}H(X-kD+\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T_{1})$
$-H(X-kD-\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T_{1})\Bigg{\\}},$ (23)

and

$\sqrt{\alpha_{2}\beta_{2}}(W^{\prime}_{II}(X)-W^{\prime}_{I}(X))=\frac{\beta_{1}}{2}\sum_{k=-\infty}^{\infty}(-1)^{k}\Bigg{\\{}\delta(X-kD+\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T_{1})$
$+\delta(X-kD-\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T_{1})\Bigg{\\}}.$ (24)

When $T_{1}<T<T_{1}+T_{2}$ and $|X|<D/2$, direct integration leads to the
following representation of the solution

$V(X,T)=W_{I}(X-\sqrt{\alpha_{2}/\beta_{2}}\tau)+W_{II}(X-\sqrt{\alpha_{2}/\beta_{2}}\tau)$
$=\frac{1}{4}\sqrt{\frac{\beta_{1}}{\alpha_{1}}}\Bigg{\\{}\sum_{k=-\infty}^{\infty}(-1)^{k}\Big{[}H(X-\sqrt{\alpha_{2}/\beta_{2}}\tau-
kD+\sqrt{\alpha_{1}/\beta_{1}}T_{1})$ $-H(X-\sqrt{\alpha_{2}/\beta_{2}}\tau-
kD-\sqrt{\alpha_{1}/\beta_{1}}T_{1})+H(X+\sqrt{\alpha_{2}/\beta_{2}}\tau-
kD+\sqrt{\alpha_{1}/\beta_{1}}T_{1})$ $-H(X+\sqrt{\alpha_{2}/\beta_{2}}\tau-
kD-\sqrt{\alpha_{1}/\beta_{1}}T_{1})\Big{]}$
$+\sqrt{\frac{\alpha_{1}\beta_{1}}{\alpha_{2}\beta_{2}}}\sum_{k=-\infty}^{\infty}(-1)^{k}\int_{X-\sqrt{{\alpha_{2}}/{\beta_{2}}}\tau}^{X+\sqrt{{\alpha_{2}}/{\beta_{2}}}\tau}\Bigg{(}\delta(\xi-
kD+\sqrt{\alpha_{1}/\beta_{1}}T_{1})$ $+\delta(\xi-
kD-\sqrt{\alpha_{1}/\beta_{1}}T_{1})\Bigg{)}d\xi\Bigg{\\}}.$ (25)

###  4.2 The timing of the “temporal switch” versus the spatial periodicity

We choose $T_{1}$ and $T_{2}$ so that

$\sqrt{\frac{\alpha_{1}}{\beta_{1}}}T_{1}=\frac{D}{2}=\sqrt{\frac{\alpha_{2}}{\beta_{2}}}T_{2},$
(26)

i.e. the first wave reaches the boundary of the elementary cell when the
temporal switch occurs.

Following the derivation, which is similar to that in Section 4.1, we deduce
that when $T>T_{1}+T_{2}$, the field $V(X,T)$ takes the form

$V(X,T)=\frac{1}{2}\sqrt{\frac{\beta_{1}}{\alpha_{1}}}\sqrt{\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}}\sum_{k=-\infty}^{\infty}(-1)^{k}\Bigg{(}H(X-kD-\sqrt{\alpha_{1}/\beta_{1}}(T-T_{2}-T_{1}))$
$-H(X-kD-\sqrt{\alpha_{1}/\beta_{1}}(T-T_{2}-T_{1}))\Bigg{)}.$ (27)

The above equation is similar to (21) subject to the replacement of the
argument $T$ in (21) by $T-T_{2}-T_{1}$ in (27) and an additional factor
$\sqrt{\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}}$ in (27). In
particular, when $X=0$ and $T>T_{1}+T_{2}$

$V(0,T)=-\frac{1}{2}\sqrt{\frac{\beta_{1}}{\alpha_{1}}}\sqrt{\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}}\sum_{k=-\infty}^{\infty}(-1)^{k}\Bigg{\\{}H(-kD-\sqrt{\alpha_{1}/\beta_{1}}(T-T_{2}-T_{1}))$
$-H(kD-\sqrt{\alpha_{1}/\beta_{1}}(T-T_{2}-T_{1}))\Bigg{\\}}=-\sqrt{\frac{\beta_{1}}{\alpha_{1}}}\sqrt{\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}}.$
(28)

By considering a periodic semi-infinite temporal stratification with the
elementary cell of the width $T_{1}+T_{2}$, after $n$ iterations we obtain

$V(0,T)=(-1)^{n}\sqrt{\frac{\beta_{1}}{\alpha_{1}}}\Bigg{(}\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}\Bigg{)}^{n/2},$
(29)

when $n(T_{1}+T_{2})<T<(n+1)T_{1}+nT_{2}.$

The above formula (28) shows that when
$\frac{\alpha_{2}\beta_{2}}{\alpha_{1}\beta_{1}}>1$, the modulus of the
displacement at the origin $|V(0,T)|$ increases exponentially as $n\to\infty.$

##  5 Chiral interfaces

The scalar problem of vibration of an elastic string considered above will
here be extended to a vector case. Both transverse and longitudinal
displacement are included in the formulation, while the coupling process is
governed by a chiral term, present in the equation or interface transmission
conditions.

Firstly, the model of a chiral continuum will be summarised since this
continuum will be used as a chiral temporal interface. Secondly, energy
considerations will be discussed followed by the analysis of a transient
configuration of a spatial chiral interface.

###  5.1 The chiral medium

A model for a chiral medium has been introduced in stages. Firstly, gyroscopic
resonators connected to elastic beams were introduced in [20] where a
gyroscopic resonator was replaced by suitable displacement boundary conditions
replacing the resonator and describing the gyroscopic action. The linearised
framework of the gyroscopic motion was used and a small angle of nutation was
assumed. The governing equations for a one-dimensional discrete chain of such
resonators with hinged bases and connected by massless springs was discussed
in [21] and the resonators were again replaced by appropriate time dependent
displacement boundary conditions. The system of coupled governing equations in
the longitudinal and transverse displacements was homogenised in [23] leading
to a model with governing equations, written for a chiral medium, in the
vector form

$-\frac{\partial^{2}}{\partial t^{2}}{{\bf U}}+{\bf
D}\frac{\partial^{2}}{\partial x^{2}}{\bf U}+\alpha{\bf
R}\frac{\partial}{\partial t}{{\bf U}}={\bf 0},$ (30)

where ${\bf U}=(u(x,t),v(x,t))^{T}$ is a real-valued vector function,
representing the longitudinal, $u(x,t)$, and transverse, $v(x,t)$,
displacements,

${\bf D}=\mbox{diag}\\{c_{1}^{2},c_{2}^{2}\\},\leavevmode\nobreak\
\leavevmode\nobreak\ {\bf R}=\begin{pmatrix}0&1\cr-1&0\end{pmatrix},$ (31)

where $c_{1}$ and $c_{2}$ are the wave speeds for the longitudinal and
transverse waves respectively, and $\alpha$ is the (re-defined) gyricity
parameter. The third term in (30) is the coupling term, and it represents the
gyroscopic force, which is orthogonal to the velocity vector.

Although, in the linearised setting, the gyroscopic force appears to be non-
conservative, it is orthogonal to the velocity vector, and for a finite chiral
elastic rod the classical energy conservation holds, as demonstrated in the
illustrative example below.

###  5.2 Energy consideration

Here, an elementary demonstration is given, based on two examples which
include vibration of a chiral finite rod and time-harmonic vibration of a
single chiral inertial resonator.

####  5.2.1 The finite elastic chiral rod

Let ${\bf U}=(u(x,t),v(x,t))^{T}$ be a real-valued vector function,
representing the displacements, which satisfy the equations of motion (30) for
the homogenised chiral elastic rod, $x\in(0,1)$ with the boundary conditions

${\bf U}(0,t)={\bf U}(1,t)=0\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{for any admissible }t.$ (32)

Multiplying equation (30) by $\frac{\partial}{\partial t}{\bf U}^{T}$ and
integrating with respect to $x$ over the interval $(0,1)$ gives

$0=\int_{0}^{1}\Big{\\{}\frac{\partial u}{\partial
t}\frac{\partial^{2}u}{\partial t^{2}}+\frac{\partial v}{\partial
t}\frac{\partial^{2}v}{\partial t^{2}}-c_{1}^{2}\frac{\partial u}{\partial
t}\frac{\partial^{2}u}{\partial x^{2}}-c_{2}^{2}\frac{\partial v}{\partial
t}\frac{\partial^{2}v}{\partial x^{2}}\Big{\\}}dx.$ (33)

Note that $(\frac{\partial}{\partial t}{{\bf U}})^{T}{\bf
R}\frac{\partial}{\partial t}{{\bf U}}={\bf 0}.$ By integrating (33) by parts,
and using the boundary conditions (32), we obtain

$\frac{d}{dt}\Big{(}K+P\Big{)}=0,$ (34)

where $K$ and $P$ represent the kinetic and potential energies, respectively,

$K=\frac{1}{2}\int_{0}^{1}\Big{(}(\frac{\partial u}{\partial
t})^{2}+(\frac{\partial v}{\partial t})^{2}\Big{)}dx,\leavevmode\nobreak\
P=\frac{1}{2}\int_{0}^{1}\Big{(}c_{1}^{2}(\frac{\partial u}{\partial
x})^{2}+c_{2}^{2}(\frac{\partial v}{\partial x})^{2}\Big{)}dx.$ (35)

Hence, the standard conservation law holds, and the total energy $K+P$ is
time-independent.

####  5.2.2 Time-harmonic regime

In the time-harmonic regime, with radian frequency $\omega$, the energy
consideration for a finite elastic rod leads to the evaluation of the first
eigenvalue. Assuming that ${\bf U}(x,t)=\hat{{\bf U}}(x)\exp(-i\omega t),$
then

$\omega^{2}\hat{{\bf U}}(x)+{\bf D}\hat{{\bf
U}}^{\prime\prime}(x)-i\omega\alpha{\bf R}\hat{{\bf U}}(x)={\bf
0},\leavevmode\nobreak\ x\in(0,1),$ (36) $\hat{{\bf U}}(0)=\hat{{\bf
U}}(1)={\bf 0}.$ (37)

In the above equations, $\hat{{\bf U}}(x)=(\hat{u}(x),\hat{v}(x))^{T}$ is a
complex valued vector function. When the gyricity parameter $\alpha$ is zero,
the problem (36), (37) splits into two uncoupled standard eigenvalue problems
for two harmonic oscillators. Additionally, in the case when $\alpha=0,$ the
following identity holds:

$\omega=\Bigg{(}\frac{\int_{0}^{1}\Big{(}c_{1}^{2}|\hat{u}^{\prime}(x)|^{2}+c_{2}^{2}|\hat{v}^{\prime}(x)|^{2}\Big{)}dx}{\int_{0}^{1}\Big{(}|\hat{u}(x)|^{2}+|\hat{v}(x)|^{2}\Big{)}dx}\Bigg{)}^{1/2},$
(38)

where the energy integral is in the numerator of (38). Using the normalisation

$\int_{0}^{1}\Big{(}|\hat{u}(x)|^{2}+|\hat{v}(x)|^{2}\Big{)}dx=1,$ (39)

and adopting the notation $H_{0}(0,1)$ for the space of vector functions,
which satisfy (37), (39) and have the finite energy integral in (38), we have
that the first eigenvalue as

$\omega_{1}=\inf_{\hat{{\bf U}}\in
H_{0}(0,1)}\Bigg{(}{\int_{0}^{1}\Big{(}c_{1}^{2}|\hat{u}^{\prime}(x)|^{2}+c_{2}^{2}|\hat{v}^{\prime}(x)|^{2}\Big{)}dx}\Bigg{)}^{1/2}.$
(40)

When $\alpha\neq 0$ and the normalisation (39) is in place, for positive
$\omega$ the identity (38) is replaced by

$\omega=\frac{\int_{0}^{1}\Big{(}c_{1}^{2}|\hat{u}^{\prime}(x)|^{2}+c_{2}^{2}|\hat{v}^{\prime}(x)|^{2}\Big{)}dx}{\alpha\int_{0}^{1}{\it
Im}(\bar{\hat{u}}\hat{v})dx+\sqrt{\alpha^{2}\Big{(}\int_{0}^{1}{\it
Im}(\bar{\hat{u}}\hat{v})dx\Big{)}^{2}+\int_{0}^{1}\Big{(}c_{1}^{2}|\hat{u}^{\prime}(x)|^{2}+c_{2}^{2}|\hat{v}^{\prime}(x)|^{2}\Big{)}dx}}.$
(41)

###  5.3 Transient scattering on an active chiral spatial interface

Consider two one-dimensional semi-infinite elastic rods joined by a one-
dimensional chiral segment. The uncoupled longitudinal and transverse
displacements in the rods are governed by (30) with $\alpha=0$. The coupled
displacements in the chiral segment are governed by (30) with the chirality
parameter $\alpha$ being non-zero.

For simplicity, the limiting case when the chiral segment height tends to zero
will be considered, with $\alpha$ approaching infinity in such a way that the
limit is an imperfect interface.

####  5.3.1 Imperfect spatial chiral interface

Assume that the chiral interface extends over the small interval $0<x<d$,
$d\ll 1$, and the following equations and interface transmission conditions
hold

$\frac{\partial^{2}}{\partial t^{2}}{\bf U}(x,t)-\alpha{\bf
R}\frac{\partial}{\partial t}{\bf U}(x,t)-{\bf D}\frac{\partial^{2}}{\partial
x^{2}}{\bf U}={\bf 0},\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{when}\leavevmode\nobreak\ 0<x<d,$ (42) $\frac{\partial^{2}}{\partial
t^{2}}{\bf U}(x,t)-{\bf D}\frac{\partial^{2}}{\partial x^{2}}{\bf U}={\bf
0},\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{outside the
segment}\leavevmode\nobreak\ [0,d],$ (43)

and the continuity transmission conditions are set at $x=0$ and $x=d$

$\Big{[}{\bf U}\Big{]}_{x=-0}^{x=+0}={\bf 0},\leavevmode\nobreak\
\Big{[}\frac{\partial}{\partial x}{\bf U}\Big{]}_{x=-0}^{x=+0}={\bf 0},$ (44)
$\Big{[}{\bf U}\Big{]}_{x=d-0}^{x=d+0}={\bf 0},\leavevmode\nobreak\
\Big{[}\frac{\partial}{\partial x}{\bf U}\Big{]}_{x=d-0}^{x=d+0}={\bf 0}.$
(45)

Assuming that the gyricity parameter $\alpha$ is large, so that the product
$\nu:=\alpha d$ remains constant, and introducing the scaled variable
$\xi=x/d$, we can re-write the equation (42) in the form:

${\bf D}\frac{\partial^{2}}{\partial\xi^{2}}{\bf U}+d\beta{\bf
R}\frac{\partial}{\partial t}{\bf U}-d^{2}\frac{\partial^{2}}{\partial
t^{2}}{\bf U}={\bf 0},\leavevmode\nobreak\ \leavevmode\nobreak\ 0<\xi<1.$ (46)

Consider the asymptotic approximations of the form

${\bf U}={\bf U}^{(0)}(\xi,t)+d{\bf
U}^{(1)}(\xi,t)+O(d^{2}),\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{when}\leavevmode\nobreak\ 0<\xi<1,$ (47)

and

${\bf U}={\bf V}^{(0)}(x,t)+O(d),\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{when}\leavevmode\nobreak\ x<0\leavevmode\nobreak\
\mbox{or}\leavevmode\nobreak\ x>d.$ (48)

Substitution of (47) into (42), and use of the transmission conditions (44)
and (45) leads to a sequence of problems for ${\bf U}^{(0)}$ and ${\bf
U}^{(1)}$. Namely, we deduce

$\frac{\partial^{2}}{\partial\xi^{2}}{\bf U}^{(0)}={\bf
0},\leavevmode\nobreak\ \leavevmode\nobreak\ 0<\xi<1,$ (49)
$\frac{\partial}{\partial\xi}{\bf U}^{(0)}={\bf 0}\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{when}\leavevmode\nobreak\
\xi=+0\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\ \xi=1-0.$ (50)

Hence, ${\bf U}^{(0)}$ is $\xi-$independent, and

${\bf U}^{(0)}(t)=V^{(0)}(0,t)=V^{(0)}(d,t).$ (51)

Next, the vector function ${\bf U}^{(1)}$ satisfies the equation

${\bf D}\frac{\partial^{2}}{\partial\xi^{2}}{\bf U}^{(1)}(\xi,t)+\beta{\bf
R}\frac{\partial}{\partial t}{\bf U}^{(0)}(t)={\bf 0},$ (52)

and

$\frac{\partial}{\partial\xi}{\bf U}^{(1)}(0,t)=\frac{\partial}{\partial
x}{\bf V}^{(0)}(0,t),\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{\partial}{\partial\xi}{\bf U}^{(1)}(1,t)=\frac{\partial}{\partial x}{\bf
V}^{(0)}(d,t).$ (53)

When $d$ is infinitesimally small, (50)-(53) lead to the following
transmission conditions for the leading-order term ${\bf V}^{(0)}$ in (48),
across the chiral interface

$\Big{[}{\bf V}^{(0)}(x,t)\Big{]}_{x=-0}^{x=+0}={\bf 0},\leavevmode\nobreak\
{\bf D}\Big{[}\frac{\partial}{\partial x}{\bf
V}^{(0)}(x,t)\Big{]}_{x=-0}^{x=+0}=-\beta{\bf R}\frac{\partial}{\partial
t}{\bf V}^{(0)}(0,t).$ (54)

Taking into account the above asymptotic approximation, in the limit, as
$d\to+0$, one can consider the transmission problem for an imperfect spatial
chiral interface, where tractions become discontinuous, and a coupling is
observed between the longitudinal and transverse displacements:

$\displaystyle{\bf D}\frac{\partial^{2}}{\partial x^{2}}{\bf
U}-\frac{\partial^{2}}{\partial t^{2}}{\bf U}(x,t)={\bf
0},\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{when}\leavevmode\nobreak\
x\neq 0,$ (55) $\displaystyle\Big{[}{\bf U}(x,t)\Big{]}_{x=-0}^{x=+0}={\bf
0},\leavevmode\nobreak\ {\bf D}\Big{[}\frac{\partial}{\partial x}{\bf
U}(x,t)\Big{]}_{x=-0}^{x=+0}=-\beta{\bf R}\frac{\partial}{\partial t}{\bf
U}(0,t).$ (56)

We also note that equivalently, instead of (55) and (56), we can write the
equation with a delta-function term as follows:

${\bf D}\frac{\partial^{2}}{\partial x^{2}}{\bf
U}-\frac{\partial^{2}}{\partial t^{2}}{\bf U}(x,t)+\beta{\bf
R}\frac{\partial}{\partial t}{\bf U}(0,t)\delta(x)={\bf 0}.$ (57)

####  5.3.2 A model scattering problem

Assume that a step-like front of the longitudinal wave is propagating in the
positive direction of the $x-$axis

${\bf U}_{\mbox{\tiny inc}}=\begin{pmatrix}1-H(x-x_{0}-c_{1}(t-t_{0}))\cr
0\end{pmatrix},$ (58)

and at time $t=t_{0}>0$ this front meets a chiral interface, positioned at the
point $x=x_{0},$ and characterised by gyricity $\beta$, as discussed in
Section 5.3.1.

The total field includes two terms

${\bf U}(x,t)={\bf U}_{\mbox{\tiny inc}}+{\bf U}_{\mbox{\tiny sc}},$ (59)

where the incident field has only the first non-zero component $u_{\mbox{\tiny
inc}}$, which satisfies the homogeneous wave equation

$\frac{\partial^{2}}{\partial t^{2}}{u_{\mbox{\tiny
inc}}}-c_{1}^{2}\frac{\partial^{2}}{\partial x^{2}}u_{\mbox{\tiny inc}}={0},$
(60)

whereas the scattered field has only the second non-zero component
$v_{\mbox{\tiny sc}}$, corresponding to the transverse vibration, and it is
triggered by an instantaneous point force, whose magnitude is proportional to
the velocity of the longitudinal vibration but the orientation of the force is
orthogonal to the longitudinal velocity. Taking into account (57), we observe
that the scattered field is the transverse transient wave, which satisfies the
following initial value problem

$c_{2}^{2}\frac{\partial^{2}}{\partial x^{2}}v_{\mbox{\tiny
sc}}-\frac{\partial^{2}}{\partial t^{2}}{v_{\mbox{\tiny
sc}}}-\beta\frac{\partial u_{\mbox{\tiny inc}}}{\partial
t}(x_{0},t)\delta(x-x_{0})={0},$ (61)

subject to the initial conditions

$v_{\mbox{\tiny sc}}(x,0)=0,\leavevmode\nobreak\ \frac{\partial}{\partial
t}v_{\mbox{\tiny sc}}(x,0)=0,$ (62)

with $\alpha$ being the chirality parameter characterising the chiral point
scatterer. Taking into account (57) and (58), the equation (61) can be re-
written in the form

$c_{2}^{2}\frac{\partial^{2}}{\partial x^{2}}v_{\mbox{\tiny
sc}}-\frac{\partial^{2}}{\partial t^{2}}{v_{\mbox{\tiny
sc}}}-\beta\delta(t-t_{0})\delta(x-x_{0})={0}.$ (63)

Taking the Fourier transform $\hat{v}_{\mbox{\tiny
sc}}(k,t)=\int_{-\infty}^{\infty}v_{\mbox{\tiny sc}}(x,t)\exp(ikx)dx$, we
deduce

$k^{2}c_{2}^{2}\hat{v}_{\mbox{\tiny sc}}(k,t)+\frac{\partial^{2}}{\partial
t^{2}}{\hat{v}_{\mbox{\tiny sc}}}(k,t)+\beta\delta(t-t_{0})e^{ikx_{0}}={0},$
(64) $\hat{v}_{\mbox{\tiny sc}}(k,0)=0,\leavevmode\nobreak\
\frac{\partial}{\partial t}\hat{v}_{\mbox{\tiny sc}}(k,0)=0,$ (65)

and hence

$\hat{v}_{\mbox{\tiny
sc}}(k,t)=-\beta\frac{\sin(kc_{2}(t-t_{0}))}{kc_{2}}H(t-t_{0})e^{ikx_{0}}.$
(66)

Using the identity

$\int_{0}^{\infty}\frac{\sin(kq)}{k}dk=\frac{\pi}{2}\mbox{sign}(q),$ (67)

together with the inverse Fourier transform, we obtain

$v_{\mbox{\tiny
sc}}(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{v}_{\mbox{\tiny
sc}}(k,t)e^{-ikx}dk$ (68) $=-\frac{\beta}{2\pi
c_{2}}H(t-t_{0})\int_{-\infty}^{\infty}\frac{\sin(kc_{2}(t-t_{0}))}{k}e^{ik(x_{0}-x)}dk$
$=-\frac{\beta}{4c_{2}}H(t-t_{0})\Big{(}\mbox{sign}(c_{2}(t-t_{0})+x-x_{0})+\mbox{sign}(c_{2}(t-t_{0})+x_{0}-x)\Big{)}$
$=-\frac{\beta}{2c_{2}}H(c_{2}(t-t_{0})-|x-x_{0}|)$
$=-\frac{\beta}{2c_{2}}\Big{(}H(x-x_{0}+c_{2}(t-t_{0}))-H(x-x_{0}-c_{2}(t-t_{0}))\Big{)}$

The resulting scattered field represents the transverse wave, as follows

${\bf U}_{\mbox{\tiny
sc}}(x,t)=\begin{pmatrix}0\cr\frac{\beta}{2c_{2}}\Big{(}H(x-x_{0}-c_{2}(t-t_{0}))-H(x-x_{0}+c_{2}(t-t_{0}))\Big{)}\end{pmatrix}.$
(69)

We note that the above scattered field is independent of $c_{1}$. This is not
a general result, but a special feature of the model problem considered here,
where the incident field is defined via a step function and the homogeneity
properties $H(\lambda x)=H(x),\delta(\lambda
x)=\lambda^{-1}\delta(x),\lambda>0,$ have been used.

In the general case of an incident wave, the scattered field may depend on
both $c_{1}$ and $c_{2}$.

Such a chiral interface can be viewed as an active interface, as it initiates
a force, orthogonal to the orientation of the velocity of the incident field,
and the magnitude of this force depends on the gyricity parameter $\beta$. In
this case, the term “active interface” is used to emphasise that additional
energy may enter the system as a result of the interaction of the incident
wave with the chiral scatterer.

For $x<x_{0},$ the resulting reflected field ${\bf U}_{\mbox{\tiny refl}}$ has
a different polarisation compared to the incident field, and it propagates
with speed $c_{2}$ compared to the speed $c_{1}$ of the incident wave

${\bf U}_{\mbox{\tiny refl}}(x,t)={\bf U}_{\mbox{\tiny
sc}}(x,t)=\begin{pmatrix}0\cr-\frac{\beta}{2c_{2}}H(x-x_{0}+c_{2}(t-t_{0}))\end{pmatrix},\leavevmode\nobreak\
x<x_{0}.$ (70)

On the other hand, for $x>x_{0}$, the transmitted wave has both components,
representing longitudinal and transverse waves, as follows

${\bf U}_{\mbox{\tiny transm}}(x,t)={\bf U}_{\mbox{\tiny inc}}(x,t)+{\bf
U}_{\mbox{\tiny sc}}(x,t)$
$=\begin{pmatrix}1-H(x-x_{0}-c_{1}(t-t_{0}))\cr\frac{\beta}{2c_{2}}(H(x-x_{0}-c_{2}(t-t_{0}))-1)\end{pmatrix},\leavevmode\nobreak\
x>x_{0}.$ (71)

We note that the coupling between the longitudinal and transverse vibrations
is an essential feature of the chiral scatterer. In the above example, for
large values of $\beta$ the chiral transient scatterer can be interpreted as a
“switch”, which initiates a transverse wave, with amplitude controlled by
gyricity, that may be significantly larger than that of the incident
longitudinal wave.

##  6 Scattering on a temporal chiral interface

In this section, the reflection of a wave on a temporal chiral interface is
considered.

One can re-write the equations of motion (30) in the form.

$\frac{\partial}{\partial t}\Big{(}{\mbox{\boldmath${\cal M}$}}(\alpha
t)\frac{\partial}{\partial t}{\bf U}(x,t)\Big{)}-{\mbox{\boldmath${\cal
M}$}}(\alpha t){\bf D}\frac{\partial^{2}}{\partial x^{2}}{\bf U}(x,t)={\bf
0},$ (72)

where the matrix function ${\mbox{\boldmath${\cal M}$}}(\alpha t)$ is the
rotation matrix defined by

${\mbox{\boldmath${\cal M}$}}(\alpha t)=\begin{pmatrix}\cos(\alpha
t)&-\sin(\alpha t)\cr\sin(\alpha t)&\cos(\alpha t)\end{pmatrix}.$ (73)

By considering a transient process, we assume that the gyricity parameter
$\alpha$ may change instantaneously across a temporal interface $t=T_{*}$,
while the displacement and the momentum remain continuous:

$\Big{[}{\bf U}\Big{]}_{t=T_{*}-0}^{t=T_{*}+0}={\bf 0},\leavevmode\nobreak\
\leavevmode\nobreak\ \Big{[}{\mbox{\boldmath${\cal
M}$}}\frac{\partial}{\partial t}{\bf U}\Big{]}_{t=T_{*}-0}^{t=T_{*}+0}={\bf
0}.$ (74)

###  6.1 Transient solutions for small and large values of the gyricity
parameter $\alpha$

Here, illustrations will be presented for solutions of the Cauchy problems on
the infinite axis $x\in{\bf R}$ and $t>0,$ corresponding to different chiral
regimes.

The role of the chirality is in the rotational coupling between the
longitudinal and transverse displacements and this leads to wave dispersion.
In particular if $\alpha=0$, then the problem is reduced to a system of
uncoupled Cauchy problems, governed by standard D’Alembert’s solutions, which
describe non-dispersive waves propagating with the speeds $c_{1}$ and $c_{2}$
for the longitudinal and transverse vibrations respectively.

On the other hand, when $\alpha\gg 1$, the solution becomes highly oscillatory
in time $t$, compared to the spatial variation in $x$. It can be observed,
that for a finite time interval the $x-$dependence appears to be “frozen”. On
a given cross-section, the variable $x$ can be considered as a fixed
parameter.

For the purposes of numerical illustration, it is convenient to express the
governing equations (30) in dimensionless form. Firstly, the following initial
conditions will be assumed:

$u(x,0)=Lf_{1}(\frac{x}{L}),\leavevmode\nobreak\ \leavevmode\nobreak\
v(x,0)=Lf_{2}(\frac{x}{L}),\leavevmode\nobreak\ \leavevmode\nobreak\ $ (75)
$\frac{\partial u}{\partial
t}(x,0)=c_{1}g_{1}(\frac{x}{L}),\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{\partial v}{\partial t}(x,0)=c_{1}g_{2}(\frac{x}{L}).$

Here $L$ is defined as a characteristic length introduced in the initial
conditions. The functions $f_{i}$ and $g_{i}$ are dimensionless.

Introduce the dimensionless variables:

$\tilde{x}=\frac{x}{L},\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{t}=\frac{c_{1}}{L}t,\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{u}=\frac{u}{L},\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{v}=\frac{v}{L},\leavevmode\nobreak\ \leavevmode\nobreak\
\lambda=\frac{c_{2}}{c_{1}},\leavevmode\nobreak\ \leavevmode\nobreak\
\gamma=\frac{\alpha L}{c_{1}}.$ (76)

The governing equations then become (dropping the tildes
$\tilde{\leavevmode\nobreak\ }$)

$\displaystyle\frac{\partial^{2}u}{\partial
t^{2}}-\frac{\partial^{2}u}{\partial x^{2}}-\gamma\frac{\partial v}{\partial
t}$ $\displaystyle=0,$ (77) $\displaystyle\frac{\partial^{2}v}{\partial
t^{2}}-\lambda^{2}\frac{\partial^{2}v}{\partial x^{2}}+\gamma\frac{\partial
u}{\partial t}$ $\displaystyle=0,$

with initial conditions

$u(x,0)=f_{1}(x),\leavevmode\nobreak\ \leavevmode\nobreak\
v(x,0)=f_{2}(x),\leavevmode\nobreak\ \leavevmode\nobreak\ \frac{\partial
u}{\partial t}(x,0)=g_{1}(x),\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{\partial v}{\partial t}(x,0)=g_{2}(x).$ (78)

It is noted that the system (77) can be written in the matrix form, similar to
(72):

$\frac{\partial}{\partial t}\Bigg{(}{\mbox{\boldmath${\cal M}$}}(\gamma
t)\frac{\partial}{\partial t}{\bf U}(x,t)\Bigg{)}-{\mbox{\boldmath${\cal
M}$}}(\gamma t){\bf D}_{\lambda}\frac{\partial^{2}}{\partial x^{2}}{\bf
U}(x,t)={\bf 0},$ (79)

where the matrix function ${\cal M}$ is defined by (73), and ${\bf
D}_{\lambda}=\mbox{diag}(1,\lambda^{2}).$ In this case, it is assumed that
$0<\lambda<1.$

As discussed in [21], for large values of the gyricity, this problem is a
singularly perturbed one and an alternative normalisation is used in [23],
which highlights the highly oscillatory behaviour of the solutions in time.

(a) The longitudinal displacement $u(x,t)$ .

(b) The transverse displacement $v(x,t)$ .

Figure 4: Displacements for the case of very large chirality ($\gamma=100$)
near the ’stationary limit’ between $t=0$ and $t=10$. The wave speed parameter
is given by $\lambda=0.5$.

The ”stationary limit” of very large chirality is illustrated by the results
shown in Fig. 4. Equations (77) have been solved numerically and the
dimensionless initial conditions are set as

$u(x,0)=v(x,0)=e^{-x^{2}},\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{\partial u}{\partial t}(x,0)=1,\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{\partial v}{\partial t}(x,0)=1,$ (80)

with $\gamma=100$ and $\lambda=0.5$. A dimensionless time frame up to $t=5$ is
chosen and an $x$ domain is chosen to be sufficiently large to minimise
reflections. The dimensionless longitudinal and transverse displacements are
shown in Figs. 4(a) and 4(b), respectively. At this relatively large value of
$\gamma$, it may be seen that the waves remain approximately stationary in the
spatial coordinate whilst they vary harmonically in time. The analytical
solution for an infinitely large value of $\alpha$ is given in [23] and
discussed later in this paper.

An example of the case of very small chirality is shown in Fig. 5. The initial
conditions for the displacements are given in (80) but the initial velocities
are set to zero. The relative chirality parameter is given as $\gamma=0.5$ and
again $\lambda=0.5$. The longitudinal and transverse displacements are shown
in Figs. 5(a) and 5(b) respectively as functions of $x$ and $t$. It is
apparent that the initial wave profile moves to the left and right with
increasing time but the shape of that initial profile changes due to the
dispersive nature of the waves.

This situation of small chirality is much closer to that of two decoupled wave
equations. For the latter case, two Gaussian profiles move to the left and
right for both the longitudinal and transverse displacements without
dispersion. The respective wave speeds in this non-dispersive, non-chiral case
are unity and $\lambda$ (see (77) with $\gamma=0$). In the chiral case, where
the gyroscopic coupling leads to dispersion, the displacement components are
displayed in Figs. 5(a) and 5(b), and the profiles associated with non-chiral
wavefronts are also shown. The characteristic lines showing these wavefronts,
$x\pm t=\mbox{const}$ and $x\pm\lambda t=\mbox{const}$, are marked on both
Figs. 5(a) and 5(b) with dashed and dotted lines respectively. Note that all
the wave motion for both components is bounded by the characteristic
corresponding to the higher wave speed.

(a) The longitudinal displacement $u(x,t)$.

(b) The transverse displacement $v(x,t)$.

Figure 5: Longitudinal and transverse displacements between $t=0$ and $t=10$.
The relative wave speed parameter $\lambda=0.5$, and the chirality parameter
$\gamma=0.5$. The non-chiral characteristics are shown in dashed and dotted
lines.

###  6.2 A temporal chiral interface of high gyricity

We assume that at time $t=0$, the gyricity switches from $0$ to $\alpha\gg 1.$
At time $t=d>0$, the gyricity switched back to zero. In this case, we consider
the temporal interface $0<t<d$. In particular, we are interested in the values
of $\alpha$ and $d$ such that $\alpha\gg 1$, while the product $\alpha d$
remains finite.

It has been demonstrated above that for high values of $\alpha$, the
vibrations of the chiral system, discussed here, can be considered on a cross-
section, with fixed $x$, and we use a vector function ${\bf Y}(\alpha,t)$,
where the $x-$dependence is omitted.

Consider the following initial value problem

$\frac{\partial}{\partial t}\Big{(}{\mbox{\boldmath${\cal M}$}}(\alpha
t)\frac{\partial}{\partial t}{\bf Y}(\alpha,t)\Big{)}={\bf 0},$ (81) ${\bf
Y}(\alpha,0)={\bf f}=\begin{pmatrix}f_{1}\cr
f_{2}\end{pmatrix},\leavevmode\nobreak\ {\mbox{\boldmath${\cal
M}$}}(0)\frac{\partial}{\partial t}{\bf Y}(\alpha,0)=\frac{\partial}{\partial
t}{\bf Y}(\alpha,0)={\bf g}=\begin{pmatrix}g_{1}\cr g_{2}\end{pmatrix},$ (82)

with ${\bf f},{\bf g}$ being the vectors representing the initial
displacements and initial velocities at $t=0.$

The solution of (81), (82) has the form

${\bf Y}(\alpha,t)=\alpha^{-1}{\bf R}({\bf I}-{\mbox{\boldmath${\cal
M}$}}^{T}(\alpha t)){\bf g}+{\bf f},$ (83)

where ${\bf I}$ is the $2\times 2$ identity matrix. We also note that

$\frac{\partial}{\partial t}{\bf Y}(\alpha,t)={\mbox{\boldmath${\cal
M}$}}^{T}(\alpha t){\bf g},$ (84)

and hence the momentum remains constant for all positive $t$

${\mbox{\boldmath${\cal M}$}}(\alpha t)\frac{\partial}{\partial t}{\bf
Y}(\alpha,t)={\bf g}={\mbox{\boldmath${\cal M}$}}(0)\frac{\partial}{\partial
t}{\bf Y}(\alpha,0).$ (85)

It also follows from (83) that

${\bf Y}(\alpha,d)-{\bf Y}(\alpha,0)=\alpha^{-1}{\bf R}({\bf
I}-{\mbox{\boldmath${\cal M}$}}^{T}(\alpha d))\frac{\partial}{\partial t}{\bf
Y}(\alpha,0),$ (86)

which shows that the displacement ${\bf Y}$ has a discontinuity of a small
amplitude across the chiral interface.

Such an interface also provides the rotational coupling between the
longitudinal and transverse vibrations.

In particular, if $\alpha$ and $d$ are chosen in such a way that $\alpha
d=(2n-1)\pi$, where $n$ is positive integer, then (86) becomes

$\Big{[}{\bf Y}(\alpha,t)\Big{]}_{t=0}^{t=d}=2\alpha^{-1}{\bf R}{\bf g},$ (87)

whereas for the case of $\alpha d=2n\pi$ we have the ideal temporal interface,
across which both the displacement and the momentum are continuous

$\Big{[}{\bf Y}\Big{]}_{t=0}^{t=d}={\bf 0},\leavevmode\nobreak\
\leavevmode\nobreak\ \Big{[}{\mbox{\boldmath${\cal
M}$}}\frac{\partial}{\partial t}{\bf Y}\Big{]}_{t=0}^{t=d}={\bf 0}.$ (88)

We also note that, with the choice of $\alpha d=\pm\frac{\pi}{2}+2\pi n$,
where $n$ is positive integer, the equation (86) yields

$\Big{[}{\bf Y}(\alpha,t)\Big{]}_{t=0}^{t=d}=\alpha^{-1}({\bf R}\pm{\bf
I}){\bf g}.$ (89)

###  6.3 An imperfect temporal interface

Assume that outside the time-interval $t\in(T^{*},T^{*}+d)$ the gyricity
parameter $\alpha$ is zero, and hence the equation (72) has the solution ${\bf
U}(x,t)$, which describes two uncoupled waves propagating with constant speeds
$c_{1}$ (longitudinal) and $c_{2}$ (transverse), respectively, which are
classical D’Alembert’s travelling waves. We note that ${\mbox{\boldmath${\cal
M}$}}(0)={\bf I}$.

However, it is assumed that within the interval $(T^{*},T^{*}+d)$ the gyricity
parameter $\alpha$ takes a large value, and within that time-interval, on a
cross-section with fixed $x$, the solution is described in Section 6.2. The
continuity of the displacement and of the momentum (see (74)) are set at
$t=T_{*}$ and $t=T^{*}+d$.

For small $d$ and large $\alpha$, we assume that $\alpha d=(2n-1)\pi$, where
$n$ is positive integer. Then it follows from (85)–(87) that across the thin
temporal chiral interface the field ${\bf U}$ is discontinuous and the
following interface conditions hold:

$\Big{[}\frac{\partial}{\partial t}{\bf
U}(x,t)\Big{]}_{t=T_{*}}^{t=T^{*}+d}=0,\leavevmode\nobreak\ \Big{[}{\bf
U}(x,t)\Big{]}_{t=T_{*}}^{t=T_{*}+d}=2\alpha^{-1}{\bf
R}\frac{\partial}{\partial t}{\bf U}(x,T_{*}).$ (90)

The above interface conditions represent the imperfect temporal interface,
across which the displacement vector has the discontinuity. For large $\alpha$
and a finite magnitude of the velocity vector, the right-hand side in the
second interface condition (90) is small. On the other hand, when the incident
wave approaches the interface with the instantaneous velocity, which is large,
then the small coefficient $\alpha^{-1}$ may counterbalance it, and hence a
finite magnitude in the displacement jump may be observed across the imperfect
temporal interface.

##  7 Examples

Here, we consider three examples for a system of temporal chiral interfaces at
$t=T,2T,\dots,nT,\dots$, with integer $n$, combined with the Cauchy problems
for two wave equations (30), where $0<c_{2}<c_{1}=1$, and the chirality
parameter is large, i.e. $\alpha\gg 1$. In this case, the duration $d$ for
each temporal interface is infinitesimally small.

The initial velocities, at time $t=0$, for both longitudinal and transverse
orientations, are assumed to be zero:

$\frac{\partial}{\partial t}u(x,0)=0,\leavevmode\nobreak\
\frac{\partial}{\partial t}v(x,0)=0,$ (91)

while the initial values of the functions $u,v$ may be defined as required.

Here, we consider three cases:

  1. 1.

In the first case, we assume that, in addition to (91), the initial conditions
for (30) are chosen in such a way, that

$u(x,0)=0,\leavevmode\nobreak\ \leavevmode\nobreak\ v(x,0)=\Phi(x),$ (92)

where $\Phi(x)$ is a smooth even function, exponentially vanishing at
infinity. In this case, in the time interval $0<t<T$ only the transverse
component $v$ of the displacement is non-zero, whereas $u\equiv 0$.

  2. 2.

In the second case, in addition to (91), we choose the initial conditions for
(30) in such a way that

$u(x,0)=\Psi(x),\leavevmode\nobreak\ \leavevmode\nobreak\ v(x,0)=0,$ (93)

where $\Psi(x)$ is a smooth even function, exponentially vanishing at
infinity. Here, in the time interval $0<t<T$ only the longitudinal component
$u$ of the displacement is non-zero, whereas $v\equiv 0$.

  3. 3.

In the third case, both $u$ and $v$ are non-zero i.e.

$u(x,0)=\Psi(x),\leavevmode\nobreak\ \leavevmode\nobreak\ v(x,0)=\Phi(x).$
(94)

At every temporal interface $t=Tn$, there is a coupling governed by the
transmission conditions (90). This gives an additional feature in the dynamic
response of the elastic system, which incorporates a chiral temporal
interface, that couples longitudinal and transverse displacements.

###  7.1 The case of the dominant transverse displacement

For the case when the initial conditions are chosen to have the form (91),
(92), and $|\alpha|$ is a large parameter, it can be shown that the transverse
displacement $v$ becomes dominant. The coupling, which occurs at chiral
temporal interfaces, yields the solution, which includes a new pattern that
incorporates a split of the wave front and the longitudinal displacement, as
demonstrated in the closed form analytical representation below.

Figure 6: Wave split at temporal chiral interfaces. Two families of
characteristics are shown, corresponding to wave speeds $c_{1}=1$ (solid
lines) and $c_{2}=1/3$ (dashed lines). The edge wave is shown and also the
’transition layer’ where an incomplete number (less than four) of
characteristics meet an interface from below.

Using D’Alembert’s representation for the solution of the Cauchy problem on
each of the temporal intervals $(n-1)T<t<nT$, for positive integer $n$, we
deduce

$u(x,t)\equiv 0,\leavevmode\nobreak\
v(x,t)=\frac{1}{2}\Big{(}\Phi(x+c_{2}t)+\Phi(x-c_{2}t)\Big{)}\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{when}\leavevmode\nobreak\ 0<t<T,$ (95)
$\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\frac{c_{2}}{2\alpha}\Big{(}\Phi^{\prime}(x+c_{1}t-T(c_{1}-c_{2}))+\Phi^{\prime}(x-c_{1}t+T(c_{1}+c_{2}))$
$\displaystyle-\Phi^{\prime}(x+c_{1}t-T(c_{1}+c_{2}))-\Phi^{\prime}(x-c_{1}t+T(c_{1}-c_{2}))\Big{)},$
$\displaystyle v(x,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\Phi(x+c_{2}t)+\Phi(x-c_{2}t)\Big{)}\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{when}\leavevmode\nobreak\ T<t<2T,$ (96)
$\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\frac{c_{2}}{2\alpha}\Big{(}\Phi^{\prime}(x+c_{1}t-T(c_{1}-c_{2}))+\Phi^{\prime}(x-c_{1}t+T(c_{1}+c_{2}))$
$\displaystyle-\Phi^{\prime}(x+c_{1}t-T(c_{1}+c_{2}))-\Phi^{\prime}(x-c_{1}t+T(c_{1}-c_{2}))$
$\displaystyle+\Phi^{\prime}(x+c_{1}t-2T(c_{1}-c_{2}))+\Phi^{\prime}(x-c_{1}t+2T(c_{1}+c_{2}))$
$\displaystyle-\Phi^{\prime}(x+c_{1}t-2T(c_{1}+c_{2}))-\Phi^{\prime}(x-c_{1}t+2T(c_{1}-c_{2}))\Big{)},$
$\displaystyle v(x,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\Phi(x+c_{2}t)+\Phi(x-c_{2}t)\Big{)}$ (97)
$\displaystyle-\frac{c_{1}c_{2}}{2\alpha^{2}}\Big{(}\Phi^{\prime\prime}(x+c_{2}t+T(c_{1}-c_{2}))+\Phi^{\prime\prime}(x-c_{2}t+T(c_{1}+3c_{2}))$
$\displaystyle-\Phi^{\prime\prime}(x+c_{2}t-T(c_{1}+c_{2}))-\Phi^{\prime\prime}(x-c_{2}t-T(c_{1}-3c_{2}))$
$\displaystyle-\Phi^{\prime\prime}(x+c_{2}t+T(c_{1}-3c_{2}))-\Phi^{\prime\prime}(x-c_{2}t+T(c_{1}+c_{2}))$
$\displaystyle+\Phi^{\prime\prime}(x+c_{2}t-T(c_{1}+3c_{2}))+\Phi^{\prime\prime}(x-c_{2}t-T(c_{1}-c_{2}))\Big{)}\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle\mbox{when}\leavevmode\nobreak\ 2T<t<3T,$

which can be continued further, using the standard D’Alembert representation
on each of the temporal interfaces, to show that the transverse displacement
$v(x,t)=\frac{1}{2}\Big{(}\Phi(x+c_{2}t)+\Phi(x-c_{2}t)\Big{)}$ remains
dominant at all times, with the magnitude of waves associated with the
longitudinal displacement $u(x,t)$ being of order $O(|\alpha|^{-1})$.

(a) $u(x,t)$

(b) $v(x,t)$

(c) two surfaces, corresponding to $u(x,t)$ and $v(x,t)$, are plotted on the
same diagram

(d) the same surfaces, $u(x,t)$ and $v(x,t)$, as in part (c) - view from above

Figure 7: The solution of the Cauchy problem for the case 1 (dominant
transverse displacement). The following values of the parameters were used in
the computations: $c_{1}=1,c_{2}=1/3,\alpha=10,A=10.$

Numerical computations are presented here for the case when
$T=1,c_{1}=1,c_{2}=1/3$. Also, the right-hand side in the initial condition
(92) is given as $\Phi(x)=\exp(-Ax^{2}),$ with $A$ being a positive constant.
The diagram of characteristics is shown in Fig. 6 (compare with Fig. 2).

In this case, the wave propagates with the speed $c_{2}$ during the time
interval $0<t<T,$ and the equation of the characteristics is $t=|x|/c_{2}.$

At time $t=T$ the coupling, induced by the chiral interface, will lead to the
formation of two families of waves, propagating with speeds $c_{1}=1$ and
$c_{2}=1/3.$ Taking into account the wave split at the temporal interface, we
can write the equations of characteristics at the interval $T<t<2T$ as

$t=|\frac{1}{c_{2}}|x|-T|+T,\leavevmode\nobreak\ \leavevmode\nobreak\
t=\frac{1}{c_{1}}|x\pm Tc_{2}|+T.$ (98)

This process can be repeated further and the resulting diagram of
charactreristics is shown in Fig. 6. We note that this diagram does not show
the magnitude of the wave. In particular, along some characteristics the wave
magnitude may be zero. This is illustrated in the analytical representation of
the solution (95)–(97), as well as in the surface plot of the solution shown
in Fig. 7.

Figure 8: The case, when the longitudinal displacement is dominant. Two
families of characteristics are shown, corresponding to wave speeds $c_{1}=1$
(solid lines) and $c_{2}=1/3$ (dashed lines). The edge wave is shown and also
the ’transition layer’ where an incomplete number (less than four) of
characteristics meet an interface from below.

###  7.2 The case of the dominant longitudinal displacement

Here, we show the example related to the choice of (91) and (93) as the
initial conditions. The right-hand side in the first initial condition (93) is
given as $\Psi(x)=\exp(-Ax^{2}),$ with $A$ being a positive constant. As in
the previous section, we also choose $T=1,c_{1}=1,c_{2}=1/3.$ The diagram of
characteristics for this case is shown in Fig. 8, and the surface plot
representing the solution of the Cauchy problem is shown in Fig. 9. We note
that the longitudinal displacement is dominant and the coupling, which occurs
at chiral temporal interfaces, yields small wave ripples associated with the
transverse displacement.

(a) $u(x,t)$

(b) $v(x,t)$

(c) two surfaces, corresponding to $u(x,t)$ and $v(x,t)$, are plotted on the
same diagram

(d) the same surfaces, $u(x,t)$ and $v(x,t)$, as in part (c) - view from above

Figure 9: The solution of the Cauchy problem for case 2 (dominant longitudinal
displacement). The following values of the parameters were used in the
computations: $c_{1}=1,c_{2}=1/3,\alpha=10,A=10.$

###  7.3 The combined case where both longitudinal and transverse
displacements are present at the initial time

Here, the example where both displacements are non-zero at the initial time is
illustrated i.e. we choose (91), (94) as the initial conditions. The right-
hand sides in the initial conditions (94) are chosen to be
$\Phi(x)=\Psi(x)=\exp(-Ax^{2}),$ with $A$ being a positive constant. We also
choose $T=1,c_{1}=1,c_{2}=1/3.$ The diagram of characteristics for this case
is shown in Fig. 10, and the surface plot representing the solution of the
Cauchy problem is shown in Fig. 11. Both displacements $u$ and $v$ are present
at all times in this case. However, due to the presence of the chiral temporal
interfaces, the wave pattern emerges, which is consistent with the diagram of
characteristics of Fig. 10. This wave pattern also includes the “transition
layer”, adjacent to the edge wave, which will both be discussed in the next
section.

Figure 10: The combined case when both the longitudinal and transverse
displacements are present in the initial conditions. Two families of
characteristics are shown, corresponding to wave speeds $c_{1}=1$ (solid
lines) and $c_{2}=1/3$ (dashed lines). The edge wave is shown and also the
’transition layer’ where an incomplete number (less than four) of
characteristics meet an interface from below.

###  7.4 The edge wave and the transition layer

Using the notion of the edge wave introduced in Section 3.1, it can be seen
from the diagrams of characteristics, shown in Figs. 6, 8, 10, that the edge
wave in all three cases propagates with the wave speed $c_{1}$. For
suffiiciently large $\alpha$, the magnitude of the edge wave in the first
case, where the transverse wave is dominant, is of order $O(|\alpha|^{-1})$,
whereas in the remaining two cases the magnitude of the edge wave is of order
$O(1)$, as illustrated in Figs. 7, 9, 11.

With the reference to Figs. 6, 8, 10, we also identify a region adjacent to
the edge wave boundary as the ’transition layer’. In this region, at every
temporal interface an incomplete set of characteristics (less than four)
intersects the interface from below. The width of the transition layer depends
on the ratio of the wave speeds $c_{1}/c_{2}$.

We also note that if the chirality parameter $\alpha$ and the interface
thickness are chosen in such a way that $\alpha d=2\pi n$, with $n$ being
positive integer, then the ideal contact conditions (88) hold across the
interface,. In this case, the standard D’Alembert solution will be observed
and no coupling will occur at temporal interfaces.

(a) $u(x,t)$

(b) $v(x,t)$

(c) two surfaces, corresponding to $u(x,t)$ and $v(x,t)$, are plotted on the
same diagram

(d) the same surfaces, $u(x,t)$ and $v(x,t)$, as in part (c) - view from above

Figure 11: The solution of the Cauchy problem for case 3 (both longitudinal
and transverse displacements are present at all times). The following values
of the parameters were used in the computations:
$c_{1}=1,c_{2}=1/3,\alpha=10,A=10.$

##  8 Concluding remarks

As shown in Sections 2.1 and 2.2, periodic time-variation in the mass density
may lead to interesting features of solutions of wave problems. More
importantly, the model of a temporally stratified medium can be considered as
a result of the discretisation of the wave equations with time-dependent mass
density and stiffness coefficients, as in (5).

The advantage of the discretised formulation in the temporally stratified
medium is that in every temporal layer, where the coefficients of the
governing equations are time-independent, the problem can be solved and a
transmission matrix, as in Section 2.2, can be identified. Furthermore, the
process can be carried out in iterations to obtain the solution of the Cauchy
problem at any given time. Although, the diagrams of characteristics, as in
Figs. 2, 6, 8, 10, do not provide information about the wave magnitude, they
readily show possible wave patterns corresponding to selected initial
conditions as well as the parameters of temporal stratification.

Sections 2, 3, 4 describe a scalar problem of vibrations of an elastic string
with time-dependent elastic and inertial properties, with the emphasis on the
resonance blow-up as $T\to+\infty.$ The second part of the paper is focused on
a vector problem, where the longitudinal and transverse displacements of the
vibrating string are dynamically coupled.

Sections 5 and 6 introduce coupled governing equations through chirality,
together with the notion of imperfect chiral interfaces. Three examples of
Cauchy problems in temporally stratified chiral elastic systems are discussed
in detail in Section 7, and they show the wave patterns, as well the coupling
process between the longitudinal and transverse vibrations within the string.
A particularly interesting feature, displayed in Figs. 6, 8, 10, is the
presence of the transition layer adjacent to the edge wave in all three cases.
Such transition layers also appear in other field pattern geometries: see, for
example, Fig. 16 in [8].

## Acknowledgements

GWM thanks the NSF for support from the US National Science Foundation through
grant DMS-2107926. ABM, H-MN, and NVM also gratefully acknowledge support from
the NSF and the hospitality of the University of Utah where this work was
initiated.

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