In the problem of scattering from a pressure release boundary, the Helmholtz integral equation may be expressed as a Fredholm equation of the second kind with the unknown surface velocity as the independent variable. The method of successive approximations was employed by Meecham [J. Ration. Mech. Anal. 5, 323–333 (1956)] to obtain a series solution to this equation, where the first term of this series is, in fact, the unshadowed Kirchhoff approximation to the solution. Meecham attempted to formulate the region of validity of the Kirchhoff approximation by determining when the remaining terms of the series could be neglected, however, his arguments used to select ‘‘smallness’’ have been questioned. An exact solution to the integral equation for a sinusoidal boundary is employed to examine the series, term by term, for convergence; and it is found that (1) the alternating nature of the series does not bring convergence when the series diverges absolutely, and (2) the absence of any propagating side orders and reflection anomalies is not a sufficient condition for convergence. Finally, to highlight the extent of the error made when using the Kirchhoff approximation, comparisons are made between reflection coefficients computed using the exact solution and several terms of the series. It is found that in the shadowed regions, the Kirchhoff approximation is in reasonable agreement with the exact solution, even when Meecham’s series diverges.