  We consider open superstring partition function Z on the disc in
time-dependent tachyon background T= f(x_i) e^{m x_0} where the profile
function f depends on spatial coordinates. We compute Z to second order in
derivatives of f and compare the result with some previously suggested
effective actions depending only on the first derivatives of the tachyon field.
We also compute the target-space stress-energy tensor in this background and
demonstrate its conservation in the ``on-shell'' case of the linear profile f=
f_0 + q_i x_i corresponding to a marginal perturbation. We comment on the role
of the rolling tachyon with linear spatial profile in the decay of an unstable
D-brane.
