set $X_{1}=0$ at the beginning of a certain $a$ phase, and denote its number by $n=0$, such that even and odd values of $n$ correspond to layers made of phase $a$ and $b$, respectively. At the initial state, the laminate is subjected to a combination of axial and transverse displacement fields as functions of $X_{1}$. Assuming that the laminate is uniform and infinite in $X_{2}$, the resultant motion $\Curx=\map\left(\Refx,t\right)$ is defined by

$$x_{1}=X_{1}+\dis 1\left(X_{1},t\right),\;x_{2}=X_{2}+\dis 2\left(X_{1},t\right),\label{eq:deformation}$$

and maps material points from the reference coordinate $\Refx$ to the current coordinate $\Curx$ (Fig.$\ $\[fig:Illustration\]). The problem amounts to determining the displacements $u_{i}\left(X_{1},t>0\right)$ for a given initial state. This is carried out by solving the balance equations for the corresponding first Piola-Kirchhoff stress tensor $\piola=\grad_{\defgrad}\hyper$, where $\defgrad=\grad_{\text{\ensuremath{\Refx}}}\map$ is the deformation gradient. When restricting attention to smooth waves[^3], the balance equations are

$$\Refdiv{\mathbf{\piola}}=\refden\deformation_{,tt},\label{eq:eom}$$

where we recall that $\refden$ is a periodic function of $X_{1}$, and it is tacitly assumed that $\map$ is continuous and twice differentiable, except at the interfaces between phases. For the given mapping , Eq.$\ $ is reduced to

$$\begin{aligned}\piolacomp{11,1}=\refden\derivativesec{u_{1}}t,\ \piolacomp{21,1}=\refden\derivativesec{u_{2}}t.\end{aligned}
\label{eq:motion}$$

In order to reformulate Eq.$\ $ in a more convenient form for the forthcoming analysis, we first introduce the Lagrangian 