  Elastic waves scattering off a periodic single and double array of thin
cylindrical defects is considered for isotropic materials. An analytical
expression for the scattering matrix is obtained by means of the
Lippmann-Schwinger formalism and analyzed in the long wavelength limit using
Schloemilch series in order to obtain explicit expressions for the poles of the
scattering matrix. The latter is then used to prove that for a specific curve
in the space of physical and geometric parameters, the scattering is dominated
by resonances, and the width of the resonances in the shear mode parallel to
the cylinders has a global minimum in parameter space. This a feature is not
observed in similar photonic or acoustic systems. The resonances in shear and
compression modes that are coupled in the plane perpendicular to the cylinders
due to the normal traction boundary condition are studied for the double array.
The analytical dependence of the width of these resonances on physical and
geometrical parameters is exploited to prove the existence of resonances with
the vanishing width, known as Bound States in the Continuum (BSC). Spectral
characteristics of BSC are explicitly found in terms of the Bloch phase and
group velocities of elastic modes.
