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In this work, an asymptotic expansion is presented that takes into account a naturally-occurring perturbation parameter in the context of a circular tube with an open-open endpoint configuration. This approach is shown to produce accurate predictions of pressure mode shapes and frequencies for arbitrary temperature distributions that mimic a wide variety of flow heating arrangements including, but not limited to, those associated with a Rijke tube. The underlying formulation consists of two linearly coupled partial differential equations that can be solved simultaneously while using a Green’s function to capture the thermoacoustic pressure. In the present investigation, the strategy leading to an accurate prediction of the unsteady pressure oscillations is fully detailed and then applied to several representative cases. Results pertaining to the pressure oscillations are systematically discussed and compared to other recently developed models in the literature.