The coupling of mass diffusion in a solution (or a melt) and the kinetic process on the solidification interface is described by a linear diffusion equation with the appropriate linear boundary conditions (J. Crystal Growth 8 (1971) 79). In the present paper, the instability and the wave-like solution of the linear problem are discussed after obtaining a full analytical solution to the governing equations and conditions. The solution is obtained by superposing a non-steady-state solution on a steady-state solution, as given in (J. Crystal Growth 8 (1971) 79). Results from the completely solved linear problem presented in this paper are different from those obtained by using only the conditions on the interface. (C) 2002 Elsevier Science B.V. All rights reserved.