= Plane-wave string mode frequency
{title2=$\omega_n=\sqrt{(2\pi n/L)^2+m^2\kappa^2}$}
For a <closed string> in a <string sigma model in an isotropic plane wave>, insert $e^{-i\omega_n\tau+2\pi in\sigma/L}$ into the transverse <wave equation>. Periodicity requires integer $n$, giving the displayed <dispersion relation>. The zero spatial mode is an oscillator of frequency $|m\kappa|$; when this mass vanishes it becomes a free particle mode and needs the position-plus-linear-time expansion.
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