  We study the superheating field of a bulk superconductor within
Ginzburg-Landau theory, which is valid near the critical temperature. We
calculate, as functions of the Ginzburg-Landau parameter $\kappa$, the
superheating field $\Hsh$ and the critical momentum $k_c$ characterizing the
wavelength of the instability of the Meissner state to flux penetration. By
mapping the two-dimensional linear stability theory into a one-dimensional
eigenfunction problem for an ordinary differential equation, we solve the
problem numerically. We demonstrate agreement between the numerics and
analytics, and show convergence to the known results at both small and large
$\kappa$. We discuss the implications of the results for superconducting RF
cavities used in particle accelerators.
