In this paper, an analytical study has been conducted on the flow and energy transfer of an unsteady compressible oscillating flow through channels filled with porous media representing stacks in thermoacoustic engines and refrigerators. The flow in the porous material is described by the Darcy momentum equation. The thickness of the channel wall is considered to be nonzero, and the entire problem is treated as a conjugate heat transfer problem, i.e., by considering conduction heat transfer inside the channel walls. Analytical expressions for the oscillating temperature, complex Nusselt number, and energy flux density are obtained after linearizing and solving the governing differential equations with long wave, short stack, and small amplitude oscillation approximations. To verify the present study, the energy flux density expression derived in this paper is compared with the expression available in the existing thermoacoustic literature. The two expressions match quantitatively for the limiting case of infinitely large pores. For infinitely large pore limits, the Nusselt number (nondimensional heat transfer between the porous media and the channel wall) obtained in the present study also agrees quantitatively with the nonporous medium expression reported in the literature. The present study indicates that refrigeration performance comparable to that of a traditional plastic parallel plate stack is achievable using reticulated vitreous carbon foam (, ) as a porous medium, which is also supported by other researchers. The system of equations developed in the present study is a helpful tool for thermal engineers and physicists to design porous stacks for thermoacoustic devices.