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We propose a reconstruction method to extract the variable mass density fromthe elastic farfields, 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 in $\Omega$ and the Lam\'eparameters are known constants. The injected small/dense inclusion, $D:=z +aB\, (\subset\subset \Omega)$ with $z$ as its location, $a\ll 1$ as its maximumradius and $B$ of unit volume, generates a sequences of resonant frequencies.These special frequencies are related to the eigenvalues of the Lam\'e volumeintegral operator defined on the domain of the inclusion and thus are, inprinciple, computable. After injecting the small inclusion at a location point$z$, we send an elastic incident plane wave at an incident frequency close toone 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 formulathat 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 totalfield is generated in the absence of the inclusion. Then we repeat theexperiment by injecting more inclusions inside $\Omega$. Using thisreconstructed field in the Lam\'e PDE system, via a numerical differentiation,we recover the values of the mass density inside $\Omega$. It is worthmentioning that, we use measurements of dimension 3 to recover a function of 3dimensions freedom. This makes the inverse problem not over determined. Inaddition, we use only the pressure wave and the $\mathtt{p}$-part of thefarfield, for the reconstruction. To our best knowledge, this is the firstresult using only one type of elastic waves for the parameter identification.

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