Extraction of the mass density using only the ${\mathtt{p}}$-parts of the elastic fields generated by injected highly dense small inclusions

Abstract

We propose a reconstruction method to extract the variable mass density from the elastic farfields, with a single incident direction, measured before and after injecting highly dense small scaled inclusions. We take as a model, the Lam\'e system where the mass density is the unknown in $\Omega$ and the Lam\'e parameters are known constants. The injected small/dense inclusion, $D:=z +a B\, (\subset\subset \Omega)$ with $z$ as its location, $a\ll 1$ as its maximum radius and $B$ of unit volume, generates a sequences of resonant frequencies. These special frequencies are related to the eigenvalues of the Lam\'e volume integral operator defined on the domain of the inclusion and thus are, in principle, computable. After injecting the small inclusion at a location point $z$, we send an elastic incident plane wave at an incident frequency close to one of the mentioned resonant frequencies, say $\omega_{n_0}$. Contrasting the ($\mathtt{p}$-parts of the) farfields generated, at one incident direction, before and after injecting this small inclusion, we provide an explicit formula that allows us to recover the total field $V^{t,\mathtt{p}}(z,-\hat{x})$ corresponding to $\mathtt{p}$-incident waves at the location $z$. This total field is generated in the absence of the inclusion. Then we repeat the experiment by injecting more inclusions inside $\Omega$. Using this reconstructed field in the Lam\'e PDE system, via a numerical differentiation, we recover the values of the mass density inside $\Omega$. It is worth mentioning that, we use measurements of dimension 3 to recover a function of 3 dimensions freedom. This makes the inverse problem not over determined. In addition, we use only the pressure wave and the $\mathtt{p}$-part of the farfield, for the reconstruction. To our best knowledge, this is the first result using only one type of elastic waves for the parameter identification.