  We present a comparison between the performance of solvers based on Nystr\"om
discretizations of several well-posed boundary integral equation formulations
of Helmholtz transmission problems in two-dimensional Lipschitz domains.
Specifically, we focus on the following four classes of boundary integral
formulations of Helmholtz transmission problems (1) the classical first kind
integral equations for transmission problems, (2) the classical second kind
integral equations for transmission problems, (3) the {\em single} integral
equation formulations, and (4) certain direct counterparts of recently
introduced Generalized Combined Source Integral Equations. The former two
formulations were the only formulations whose well-posedness in Lipschitz
domains was rigorously established. We establish the well-posedness of the
latter two formulations in appropriate functional spaces of boundary traces of
solutions of transmission Helmholtz problems in Lipschitz domains. We give
ample numerical evidence that Nystr\"om solvers based on formulations (3) and
(4) are computationally more advantageous than solvers based on the classical
formulations (1) and (2), especially in the case of high-contrast transmission
problems at high frequencies.
