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.

14 pages, 4 figures, accepted for Publication in the Astrophysical Journal. Primary changes: added discussion of wave-action conservation as well as a section describing the energy and angular momentum transport properties of the solutions