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We propose a reconstruction method to extract the variable mass density fromthe elastic far-fields, with a single incident direction, measured before andafter injecting highly dense small scaled inclusions. We take as a model, theLam\'e system where the mass density is the unknown and the Lam\'e parametersare 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 unitvolume, generates a sequences of resonant frequencies. These specialfrequencies are related to the eigenvalues of the Lam\'e volume integraloperator defined on the domain of the inclusion. Therefore, these resonantfrequencies are, in principle, computable. After injecting the small inclusionat a location point $z$, we send an elastic incident plane wave at an incidentfrequency 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 formulathat 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 ofthe medium, in the absence of the inclusion, and $e_{n_0}^B(\cdot)$ are theeigenfunctions corresponding to the chosen eigenvalue (which is not necessarilya simple eigenvalue) of the Lam\'e volume integral operator stated on thescaled 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'slocation $z$ inside $\Omega$. Finally, using this reconstructed field in theLam\'e PDE system, via a numerical differentiation, we recover the values ofthe mass density inside $\Omega$.

Extraction of the mass density using elastic fields generated by injected highly dense small scaled inclusions