On one-dimensional vibrating systems

Abstract
An exact expression is derived for the frequency equation of a linear vibrating system with arbitrary masses. By considering the particular case in which all the masses are equal but for a few isolated exceptions, the properties of isotopic mass defects in a homogeneous one-dimensional chain are simply deduced. A stochastic model in which the masses follow a given probability distribution is then discussed, and following a method introduced by Weiss & Maradudin (1958) expansions for the spectrum are obtained in terms of moments about the mean mass. Hence power series expansions are derived for the long-wave region of the spectrum, and these are extended to models in which short-range order is present. An alternative formulation offers reasonable hope of calculating spectra over the whole frequency range. Finally, a number of general properties of vibration band spectra in one dimension are obtained.

This publication has 4 references indexed in Scilit: