The Brownian Motion of Strings and Elastic Rods

Abstract

The method introduced by Ornstein is applied to calculate the Brownian-motion mean-square deviation for strings and for elastic rods, the surrounding medium being a gas. For the string, a varying tension and elastic binding at the ends are supposed, and a formula is obtained for the mean-square deviation of any point at time $t$, having started with a given deviation of that point; the result contains infinite series. This result is specialized to the string with fixed endsand constant tension. For the mid-point, and for a limited time interval, the series are summed; for $t\ensuremath{\rightarrow}\ensuremath{\infty}$, the result is given for all points, agreeing with that given by Ornstein for the mid-point. Elastic rods are treated similarly, and similar results are obtained. The effect of gravity, when the rod is vertical, is introduced by a simple and consequent preturbation method, and a formula is obtained for the mean-square deviation of the lower end; this agrees closely with Houdijk's experimental results. The time dependence given by the complete formula cannot yet be tested, for Houdijk gives only long-time mean values in his publication.

Keywords

Affiliated Institutions

• University of Michigan US

Related Publications