 Eigenfrequency

An eigenfrequency is the natural frequency associated with a free normal mode. Always specify whether it means the angular frequency $\omega$ or the frequency $f=\omega /(2\pi )$ in hertz. For a uniform fixed-end wave equation on a string, ${\omega }_n=n\pi \sqrt{\tau /\mu }/L$ and $f_n=n\sqrt{\tau /\mu }/(2L)$; the amplitude does not change either within the linear model. In a generalized eigenvalue problem for small oscillations, the eigenvalue is ${\omega }^2$, rather than $\omega$ itself.

For a first-order system $\dot{z}=iAz$, a real eigenvalue $\lambda$ of $A$ instead gives a signed eigenfrequency in $e^{i\lambda t}$. A positive imaginary part of $\lambda$ gives exponential decay with this sign convention. Thus a complex damped eigenfrequency encodes both oscillation and decay; it is not just the number of cycles per second.