  We study the transverse mode instability (TMI) in the limit where a single
higher-order mode (HOM) is present. We demonstrate that when the beat length
between the fundamental mode and the HOM is small compared to the length scales
on which the pump amplitude and the optical mode amplitudes vary, TMI is a
three-wave mixing process in which the two optical modes beat with the
phase-matched component of the index of refraction that is induced by the
thermal grating. This limit is the usual limit in applications, and in this
limit TMI is identified as a stimulated thermal Rayleigh scattering (STRS)
process. We demonstrate that a phase-matched model that is based on the
three-wave mixing equations can have a large computational advantage over
current coupled mode methods that must use longitudinal step sizes that are
small compared to the beat length.
