Revealing the dispersion of phonons from stochastic excitation of wave propagation
Résumé
The band structure is a central concept in the field of phononics. Indeed, capturing the dispersion of Bloch waves gives invaluable information on allowed propagation modes, their velocity, the existence of local resonances, and the occurrence of band gaps. A band structure are usually obtained by solving an eigenvalue problem that is defined on a closed and bounded domain, resulting in a discrete spectrum. As the wavenumber is varied continuously, the eigenvalues form bands in the dispersion relation. There are at least two cases, however, that resist reduction to a linear eigenvalue problem: first, when dispersive material loss is taken into account and second, when the unit-cell of the crystal extends beyond any bound, as in the case of phononic crystal of holes or pillars on a semi-infinite substrate. We introduce a technique to obtain the phononic band structure that does not rely on searching for eigenvalues, but instead produces a mapping of the resolvent set in dispersion space. In spectral theory, indeed, the spectrum is the singular complement of the resolvent set in the complex plane. The idea is then to obtain the resolvent set of the dynamical Helmholtz equation as a function of wavenumber. The method has been implemented with finite element analysis and has been applied to several problems in phononic crystal theory. In the case of dispersive loss, the complex poles of the density of states give a direct account of propagation loss of each dispersion branch as a function of frequency and wavenumber. In the case of phononic crystals of finite-depth holes or of finite-height pillars sitting on a semi-infinite substrate, the dispersion inside the sound cone - or radiative region - is obtained and leaky guided waves can be identified.