An analytical solution of the countercurrent thermal regenerator problem when its cyclic steady-state is established is given. It takes account of the flushing phase, i.e. of that part of the blow involved in removing the residue of fluid from a previous blow. In fact, it is possible for some gas particles to remain always within the regenerator, without exiting either end. The proposed analytical approach is based on a Lagrange frame of reference for evaluating the fluid temperature during an entire operation cycle. This temperature is obtained as a function of the time and location within the regenerator and its profile depends on two dimensionless parameters: the reduced length and the flushing ratio. The method allows the matrix temperature profile to be obtained, too. The knowledge of the gas and matrix dimensionless temperature profiles in a graphical form allows a complete estimation of the dimensionless heat exchanged between gas and matrix during a semi-period at a given location.