J. Phys. I France
Volume 3, Numéro 6, June 1993
Page(s) 1259 - 1280
DOI: 10.1051/jp1:1993182
J. Phys. I France 3 (1993) 1259-1280

A "wave automaton" for wave propagation in the time domain: I. Periodic systems

Patrick Sebbah, Didier Sornette and Christian Vanneste

Laboratoire de Physique de la Matière Condensée CNRS URA 190, Université de Nice-Sophia Antipolis, Parc Valrose, B.P. 71, 06108 Nice Cedex 2, France

(Received 22 December 1992, accepted 9 February 1993)

In Vanneste et al., Europhys. Lett. 17 (1992) 715, a new lattice model was introduced for the dynamical propagation of waves in arbitrary heterogeneous media, which is efficient for calculations on large systems ( $1\,024 \times 1\,024$) over long times (several 10 6 inverse band widths). Instead of starting from a wave equation or a Hamiltonian which needs to be discretized for numerical implementation, the model is defined by the set of S-matrices, one for each node, describing the interaction of the wave field with the scatterers. Here, we expose in detail the general method of construction of the S-matrices and discuss the physical meaning of the dynamical S-matrix approach. We calculate the properties of this model for a class of parameters in the periodic case and exhibit the form of the Bloch modes, the dispersion relation and the mode density. In a companion paper, we study the transport of wave packets in arbitrary random media.

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