Herglotz wave functions play an important role in a class of reconstruction methods for inverse scattering problems known as linear sampling methods. We here consider these functions in the setting of linearized elasticity and derive representations in terms of eigenfunctions to the Navier operator in two spatial dimensions. We then show the important property that the elastic Herglotz Wave functions are dense in the space of solutions to the Navier equation with respect to the [H ¹ (D)]² norm for any bounded Lipschitz domain D. The proof of this property in three-dimensions, not essentially different from the 2D argument, is also outlined. The paper is concluded with an application of the approximation property in the mathematical foundation of the linear sampling method for the reconstruction of rigid obstacles from the knowledge of the far field operator.