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We present a theoretical stability analysis for an expanding accretion shock that does not involve a rarefaction wave behind it. The dispersion equation that determines the eigenvalues of the problem and the explicit formulae for the corresponding eigenfunction profiles are presented for an arbitrary equation of state and finite-strength shocks. For spherically and cylindrically expanding steady shock waves, we demonstrate the possibility of instability in a literal sense, a power-law growth of shock-front perturbations with time, in the range of hc h 1+2M2 , where h is the D'yakov-Kontorovich parameter, hc is its critical value corresponding to the onset of the instability and M2 is the downstream Mach number. Shock divergence is a stabilizing factor and, therefore, instability is found for high angular mode numbers. As the parameter h increases from hc to 1+2M2 , the instability power index grows from zero to infinity. This result contrasts with the classic theory applicable to planar isolated shocks, which predicts spontaneous acoustic emission associated with constant-amplitude oscillations of the perturbed shock in the range hc h 1+2M2 . Examples are given for three different equations of state: ideal gas, van der Waals gas and three-terms constitutive equation for simple metals.