Abstract
The asymptotic distribution of the eigenvalues ${k}_{n}$ of the scalar wave equation $\ensuremath{\Delta}u+{k}^{2}u=0$ is calculated for a three-dimensional finite domain of general cylindrical shape with Dirichlet and Neumann boundary conditions. In the limit of large $k$, the number $\overline{N}(k)$ of modes not exceeding $k$, smoothed in order to eliminate its fluctuating part, is determined. Four terms of the expansion of $\overline{N}(k)$ are obtained. The boundary effects for the thermal phonon radiation of thin films are discussed as an application. The respective results for the electromagnetic vector waves in a lossless closed cavity (blackbody radiation) are presented as well. Here the constant term in the expansion is independent of the shape of the domain and does not account for corners or connectivity. $E$- and $H$-type resonances have to be observed separately in order to yield the complete shape dependence of $\overline{N}(k)$. The edge- and curvature-dependent corrections of the Planck, Wien, and Stefan-Boltzmann radiation formulas are given.
Published Version
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