  We apply the stationary phase method developed in (Assier, Shanin \&
Korolkov, QJMAM, 76(1), 2022) to the problem of wave diffraction by a
quarter-plane. The wave field is written as a double Fourier transform of an
unknown spectral function. We make use of the analytical continuation results
of (Assier \& Shanin, QJMAM, 72(1), 2018) to uncover the singularity structure
of this spectral function. This allows us to provide a closed-form far-field
asymptotic expansion of the field by estimating the double Fourier integral
near some special points of the spectral function. All the known results on the
far-field asymptotics of the quarter-plane problem are recovered, and new
mathematical expressions are derived for the secondary diffracted waves in the
plane of the scatterer.
