  Parker's analytic Cartesian interface dynamo is generalized to the case of a
shear layer of finite thickness and low resistivity ("tachocline"), bounded by
a perfect conductor ("radiative zone") on the one side, and by a highly
diffusive medium ("convective zone") supporting an $\alpha$-effect on the other
side. In the limit of high diffusivity contrast between the shear layer and the
diffusive medium, thought to be relevant for the Sun, a pair of exact
dispersion relations for the growth rate and frequency of dynamo modes is
analytically derived. Graphic solution of the dispersion relations displays a
somewhat unexpected, non-monotonic behaviour, the mathematical origin of which
is elucidated. The dependence of the results on the parameter values (dynamo
number and shear layer thickness) is investigated. The implications of this
result for the solar dynamo problem are discussed.
