  This paper proposes, for wave propagating in a globally perturbed half plane
with a perfectly conducting step-like surface, a sharp Sommerfeld radiation
condition (SRC) for the first time, an analytic formula of the far-field
pattern, and a high-accuracy numerical solver. We adopt the Wiener-Hopf method
to compute the Green function for a cracked half plane, a background for the
perturbed half plane. We rigorously show that the Green function asymptotically
satisfies a universal-direction SRC (uSRC) and radiates purely outgoing at
infinity. This helps to propose an implicit transparent boundary condition for
the scattered wave, by either a cylindrical incident wave due to a line source
or a plane incident wave. Then, a well-posedness theory is established via an
associated variational formulation. The theory reveals that the scattered wave,
post-subtracting a known wave field, satisfies the same uSRC so that its
far-field pattern is accessible theoretically. For a plane-wave incidence,
asymptotic analysis shows that merely subtracting reflected plane waves, due to
non-uniform heights of the step-like surface at infinity, from the scattered
wave in respective regions produces a discontinuous wave satisfying the uSRC as
well. Numerically, we adopt a previously developed perfectly-matched-layer
(PML) boundary-integral-equation method to solve the problem. Numerical results
demonstrate that the PML truncation error decays exponentially fast as
thickness or absorbing power of the PML increases, of which the convergence
relies heavily on the Green function exponentially decaying in the PML.
