  This study builds on a recent paper by Lai et al [Appl. Comput. Harmon.
Anal., 2018] in which a novel boundary integral formulation is presented for
scalar wave scattering analysis in two-dimensional layered and half-spaces. The
seminal paper proposes a hybrid integral representation that combines the
Sommerfeld integral and layer potential to efficiently deal with the boundaries
of infinite length. In this work, we modify the integral formulation to
eliminate the fictitious eigenvalues by employing Burton-Miller's approach. We
also discuss reasonable parameter settings for the hybrid integral equation to
ensure efficient and accurate numerical solutions. Furthermore, we extend the
modified formulation for the scattering from a cavity in a half-space whose
boundary is locally perturbed. To address the cavity scattering, we introduce a
virtual boundary enclosing the cavity and couple the integral equation on it
with the hybrid equation. The effectiveness of the proposed method is
demonstrated through numerical examples.
