  I consider the nonaxisymmetric linear theory of a rotating, isothermal
magnetohydrodynamic (MHD) shear flow. The analysis is performed in the shearing
box, a local model of a thin disk, using a decomposition in terms of shearing
waves, i.e., plane waves in a frame comoving with the shear. These waves do not
have a definite frequency as in a normal mode decomposition, and numerical
integration of a coupled set of amplitude equations is required to characterize
their time dependence. Their generic time dependence, however, is oscillatory
with slowly-varying frequency and amplitude, and one can construct accurate
analytical solutions by applying the Wentzel-Kramers-Brillouin method to the
full set of amplitude equations. The solutions have the following properties:
1) Their accuracy increases with wavenumber, so that most perturbations that
fit within the disk are well-approximated as modes with time-dependent
frequencies and amplitudes. 2) They can be broadly classed as incompressive and
compressive perturbations, the former including the nonaxisymmetric extension
of magnetorotationally unstable modes, and the latter being the extension of
fast and slow modes to a differentially-rotating medium. 3) Wave action is
conserved, implying that their energy varies with frequency. 4) Their shear
stress is proportional to the slope of their frequency, so that they transport
angular momentum outward (inward) when their frequency increases (decreases).
The complete set of solutions constitutes a comprehensive linear test suite for
numerical MHD algorithms that incorporate a background shear flow. I conclude
with a brief discussion of possible astrophysical applications.
