The circular duct between the cylinder and displacer serves as a regenerator in free-piston Stirling engines. The cylinder wall is fixed and the displacer wall is in reciprocating motion during the steady operation of the engine. The basic equations of the working fluid and regenerative duct are derived using the Lagrangian method in terms of the displacement of the displacer, so that time does not appear in the equations. A relation is derived between the cylinder and displacer wall temperatures to obtain the initial wall temperature distributions. A computer program is written in FORTRAN and the governing equations, which include the pressure fluctuations due to the flow reversals, are solved numerically using a finite difference method. The results and discussion are presented. (C) 2002 Elsevier Science Ltd and IIR. All rights reserved.