A two-dimensional unsteady transonic flow of a perfect gas with constant specific heats is considered, solutions being found in the form of perturbations from a uniform, sonic, isentropic flow. Longitudinal viscous stress terms are retained so that shock waves can be included. The case where the characteristic time of a temporal flow disturbance is large compared with the time taken by a sonic disturbance to traverse the transonic regime is studied. A similarity solution involving an arbitrary function of time is employed, such that the channel walls are in general not stationary. Solutions are presented for thick (shock fills transonic region) and thin (shock tends to a discontinuity) shock waves for both decelerating and accelerating channel flows. For the thin-shock case, both numerical and asymptotic solutions are given. Flow pictures illustrating variations in shock position and structure as well as velocity distributions are shown for exponentially decreasing and for harmonic temporal flow disturbances.