A linearized perturbation about the Vialov–Nye fixed-span solution for a steady-state ice sheet yields a Sturm-Liouville problem. The numerical eigenvalue problem is solved and the resulting normal modes are used to compute Green's and influence functions for perturbations to the accumulation rate, the rate factor and for long-wavelength basal topography. The eigenvalue for the slowest mode is approximately the same as that predicted by the zero-dimensional theory. It is found that the sensitivity of the steady profile to accumulation is greatest in the central area of the ice sheet, while the sensitivity to rate factor is greatest near the margin. The antisymmetric perturbation provides information about the relaxation time for divide motion and spatial variation in the sensitivity of divide deviation from the ice-sheet centre to accumulation rate variations. The use of the method for model initialization is considered. Forcing deviations of 30% give relative errors in the perturbation of about 10%.