= Eigenfrequency
{title2=$\omega$}
= Normal mode frequency
{synonym}
= Normal frequency
{synonym}
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.
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