  Analytic representation formulas and power series are developed to describe
the band structure inside periodic elastic crystals made from high contrast
inclusions. We use source free modes associated with structural spectra to
represent the solution operator of the Lam\'e system inside phononic crystals.
Convergent power series for the Bloch wave spectrum are obtained using the
representation formulas. An explicit bound on the convergence radius is given
through the structural spectra of the inclusion array and the Dirichlet spectra
of the inclusions. Sufficient conditions for the separation of spectral
branches of the dispersion relation for any fixed quasi-momentum are
identified.
