# Direct and Inverse Problems of the Theory of Wave Propagation in an Elastic Inhomogeneous Medium with Discontinuous Parameters

Aydys A. Sedipkov*

Let us consider the process of plane wave propagation in Euclidian space $\mathbb{R}^3$, which is filled with an elastic isotropic medium. The mechanic parameters of this medium depend only on one spatial coordinate $y$. We suppose that the waves are polarized along a straight line parallel to the plane $y=0$. \newline Under these conditions, the displacement points $w$ of the medium on the equilibrium depend only on the coordinates $z$ and time $t$ and satisfy the equation of acoustics $$\tag{\star} (\mu w_y)_y=\rho w_{tt},$$ where $\rho=\rho(y)$ -- density, $\mu=\mu(y)$ -- shear modulus.\newline We set \begin{align*}&-\infty=y^-_0<y^-_*\leq y^-_1<\ldots<y^-_m<y^-_{m+1}=0,\\&0=y^+_{n+1}<y^+_n<\ldots<y^+_1\leq y^+_*<y^+_0=\infty. \end{align*} It will be supposed that the functions $\rho, \mu$ are constant outside the interval $[y^-_*,y^+_*]$, strictly positive and twice continuously differentiable on the intervals $(y^+_{k+1},y^+_k)$, $k=0,...,n-1$, $(y^-_{l},y^-_{l+1})$, $l=0,...,m-1,$ and $(y^-_m,y^+_n)$.\newline We assume that the wave process is generated by non-uniform boundary conditions such as $$\tag{\diamond} w(+0,t)-w(-0,t)=g_0(t), \;\; \mu(0)w_y(+0,t)-\mu(0)w_y(-0,t)=g_1(t),$$ where functions $g_0$ and $g_1$ are equal to zero outside of the interval $(0, + \infty)$.\newline As the medium should be rest to the beginning of this influence $$\tag{\dagger} w\mid_{t\leq{0}}=0.$$ $\textbf{Direct problem:}$ Let the medium parameters $\rho(y)$, $\mu(y)$ and the functions $g_0(t)$, $g_1(t)$ be given. It is necessary to find the function $w(y,t)$ satisfying the equation $(\star)$ and the conditions $(\diamond)$, $(\dagger).$\newline $\textbf{Inverse problem:}$ Let the functions $w(+0,t)$, $w(-0,t)$, $w_y(+0,t)$, $w_y(-0,t)$ be given. It is necessary to find the medium parameters $\rho(y)$, $\mu(y)$.  For the direct problem the theorem of the existence and uniqueness is proved. Also obtained a special representation for the solution of the direct problem.\newline By using the special representation for the solution of the direct problem and reducing the inverse problem to the corresponding inverse spectral problem, the algorithm for the reconstruction of the acoustic impedance $\sigma=\sqrt{\rho\mu}$ is constructed.\newline\newline \textit{This work was supported in part by Russian Foundation for Basic Research under Grant 12-01-00390.} \begin{thebibliography}{99} \bibitem{sed1}Sedipkov~A.\ A. {\em Direct and inverse problems of the theory of wave propagation in an elastic inhomogeneous medium}, Journal of Inverse and Ill-Posed Problems, 2011, Volume 19, Issue 3. P. 511--523. \bibitem{sed2} Sedipkov~A.\ A.\ {\em The inverse spectral problem for the impedance equation with piecewise continuous coefficients}, Proceeding of 2011 World congress on Engineering and Technology. IEEE press. Shanghai, 2011, Vol.~1. P.~509--511. \bibitem{sed3} Sedipkov~A.\ A.\ {\em The inverse spectral problem for the Sturmâ€“-Liouville operator with discontinuous potential}, Journal of Inverse and Ill-Posed Problems, 2012, Volume 20, Issue 2, P.~139--167. \end{thebibliography}

Mathematics Subject Classification: 35L53 65F18

Keywords: elastic medium; acoustic impedance; jost function; inverse spectral problem;

Minisymposion: Inverse Problems in Elasticity