A thin annular plate is subjected to a uniform tensile field at its inner edge which leads to compressive circumferential stresses. When the intensity of the applied field is strong enough, elastic buckling occurs circumferentially, leading to a wrinkling pattern. Using a linear non-homogeneous pre-bifurcation state, the linearised eigenvalue problem describing this instability is cast as a fourth-order linear differential equation with variable coefficients. This problem is investigated numerically and it is shown that the simple application of the Galerkin technique reported in the literature leads to gross errors in the corresponding approximations. Several novel mathematical features of the eigenvalue problem are included as well.