  This paper proposes well-conditioned boundary integral equations based on the
Burton-Miller method for solving transmission problems. The Burton-Miller
method is widely accepted as a highly accurate numerical method based on the
boundary integral equation for solving exterior wave problems. While this
method can also be applied to solve the transmission problems, a
straightforward formulation may yield ill-conditioned integral equations.
Consequently, the resulting linear algebraic equations may involve a
coefficient matrix with a huge condition number, leading to poor convergence of
Krylov-based linear solvers. To address this challenge, our study enhances
Burton-Miller-type boundary integral equations tailored for transmission
problems by exploiting the Calderon formula. In cases where a single material
exists in an unbounded host medium, we demonstrate the formulation of the
boundary integral equation such that the underlying integral operator ${\cal
A}$ is spectrally well-conditioned. Specifically, ${\cal A}$ can be designed in
a simple manner that ensures ${\cal A}^2$ is bounded and has only a single
eigenvalue accumulation point. Furthermore, we extend our analysis to the
multi-material case, proving that the square of the proposed operator has only
a few eigenvalues except for a compact perturbation. Through numerical examples
of several benchmark problems, we illustrate that our formulation reduces the
iteration number required by iterative linear solvers, even in the presence of
material junction points; locations where three or more sub-domains meet on the
boundary.
