This paper concerns itself with how the cut-off frequencies and modal shapes of cylindrical acoustic waveguides are altered by the presence of temperature gradients induced by an applied temperature distribution along the duct walls. A physical model is first formulated which incorporates an inhomogeneous sound speed as well as a density gradient. The associated mathematical model is then a generalized eigenproblem. This is then discretized via the control region approximation (a finite difference scheme) yielding a generalized matrix eigenproblem. Under the assumption that the boundary temperature distribution is nearly constant, we apply a procedure of Schrödinger to express the propagation constants and modal functions as perturbation series whose successive terms can be generated recursively. The case of a rectangular duct with temperature variations in the cross section is considered in detail. All numerical computations are performed usingMATLAB©.