  We propose a reconstruction method to extract the variable mass density from
the elastic far-fields, 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 and the Lam\'e parameters
are kept as known constants. The injected small/dense inclusion, $D:=z +a B$
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. Therefore, these resonant
frequencies 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 farfields generated, at one incident direction,
before and after injecting this small inclusion, we provide an explicit formula
that allows us to recover the quantity $G^{\omega}(z,x)\cdot
\int_{B}e_{n_0}^B(\eta)\,d\eta$ where $G^\omega(z,x)$ is the Green's tensor of
the medium, in the absence of the inclusion, and $e_{n_0}^B(\cdot)$ are the
eigenfunctions corresponding to the chosen eigenvalue (which is not necessarily
a simple eigenvalue) of the Lam\'e volume integral operator stated on the
scaled domain $B$. Here $x$ is outside of the domain of interest $\Omega$ and
$z \in \Omega$. Then we repeat the experiment by moving the inclusion's
location $z$ inside $\Omega$. Finally, using this reconstructed field in the
Lam\'e PDE system, via a numerical differentiation, we recover the values of
the mass density inside $\Omega$.

Analysis of PDEs