Energy density and flow distribution in a vibrating string

Abstract

The kinetic and potential energy of a vibrating string is considered in the first-order approximation of purely transverse small amplitude linear oscillations. The energy continuity equation is obtained for an energy density having a potential that depends on second-order spatial derivatives of the perturbation. The concomitant flow involves two terms $$-\partial _{t}\psi \partial _{z}\psi$$ and $$\psi \partial _{z}\partial _{t}\psi$$ that will be shown to correspond to the kinetic and potential energy flows respectively. The string’s transverse force is consistent with its derivation from minus the gradient of this potential. In contrast, the widely accepted potential energy of a string depends on the perturbation’s first-order derivative squared. However, this expression does not yield the appropriate force required to satisfy a wave equation. The potential energy controversy is thus resolved in favor of the $$\varrho _{\text {pot}}=-\frac{1}{2}T\psi \partial _{z}^{2}\psi$$ potential function. Contrary to what is usually recognized, the potential energy spatial distribution is shown to be uniquely determined. These results have far-reaching consequences pertaining the wave energy distribution in other mechanical systems.