Spectrum of the free rod under tension and compression

In this paper, we study the spectrum of the one-dimensional vibrating free rod equation $u^{(4)}-\tau u\mathrm{"}=\mu u$ under tension $(\tau >0)$ or compression $(\tau <0)$. The eigenvalues $\mu$ as functions of the tension/compression parameter $\tau$ exhibit three distinct types of behavior. In particular, eigenvalue branches in the lower half-plane exhibit a cascading pattern of barely-avoided crossings. We provide a complete description of the eigenfunctions and eigenvalues by implicitly parameterizing the eigenvalue curves. We also establish properties of the eigenvalue curves such as monotonicity, crossings, asymptotic growth, cascading and phantom spectral lines.