An exact periodic solution of the unsteady energy equation for an incompressible fluid with constant properties is derived to illustrate the effect of an oscillation through the viscous dissipation on a temperature field. The flow field used here is a generalization of the well-known Couette flow solution of steady flow, in which one wall is at rest and the other wall oscillates in its own plane about a constant mean velocity. The solution is subject to two boundary conditions that correspond to the heat-transfer and thermometer problems. In order to have some suggestions about the approximate solutions, the solution is compared with its own approximate form. The temperature field consists of a time-mean, first and second harmonic fluctuation. The time-mean temperature profiles show the large influence of oscillation. The time-mean heat flux into or the time-mean temperature of the oscillating wall increases with frequency, and is ultimately proportional to the square root of the frequency. In § 4 the present exact solution of the Couette flow is compared with the formerly obtained approximate solution of the flat plate boundary-layer flow in terms of the wall characteristic values at high frequencies.