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Classroom Notes: Disordered Lattices: Normal Modes and Localization

J. A. Scales

Jack K. Cohen
Organization: Center for Wave Phenomena, Colorado School of Mines
Journal / Anthology

The Mathematica Journal
Year: 1996
Volume: 5
Issue: 3
Page range: 68-72

Classical waves propagating in random or other disordered media have their paths distorted and are dispersed by scattering from heterogeneities. This can result in complete trapping of waves, with all the energy being converted into localized random fluctuations. AT any length scale, elastic waves can be distorted by scattering from heterogeneities that are small compared to the wavelength; this results in dispersion, attenuation, and anisotropy. A good example of a disordered medium is the earth. In seismology, the wavelengths are long compared with the fine geologic layering. In the laboratory, ultrasonic wavelengths are long compared to the pores and cracks within an individual rock sample.

A simple model of this "localization" of energy can be studied via Mathematica by replacing the continuum mechanical equations of elastic wave propagation by a discrete dynamical system of masses connected by springs. When energy traveling down this lattice encounters an isolated heterogeneity, it can excite vibrational modes that are exponentially damped about the location of the heterogeneity. Localization, then, is the cumulative result of this tendency of disorder to convert propagating energy into random fluctuations. In this note we study elastic localization in the context of the one-dimensional disordered lattice. We will investigate the normal modes of discrete dynamical systems and see directly how localization arises by perturbing the spring constants and masses.

*Science > Physics > Wave Motion