  In this paper we prove that for stable semi-discretizations of the wave
equation for the WaveHoltz iteration is guaranteed to converge to an
approximate solution of the corresponding frequency domain problem, if it
exists. We show that for certain classes of frequency domain problems, the
WaveHoltz iteration without acceleration converges in $O({\omega})$ iterations
with the constant factor depending logarithmically on the desired tolerance. We
conjecture that the Helmholtz problem in open domains with no trapping waves is
one such class of problems and we provide numerical examples in one and two
dimensions using finite differences and discontinuous Galerkin discretizations
which demonstrate these converge results.
