  In this paper we extend analysis of the WaveHoltz iteration -- a time-domain
iterative method for the solution of the Helmholtz equation. We expand the
previous analysis of energy conserving problems and prove convergence of the
WaveHoltz iteration for problems with impedance boundary conditions in a single
spatial dimension. We then consider interior Dirichlet/Neumann problems with
damping in any spatial dimension, and show that for a sufficient level of
damping the WaveHoltz iteration converges in a number of iteration independent
of the frequency. Finally, we present a discrete analysis of the WaveHoltz
iteration for a family of higher order time-stepping schemes. We show that the
fixed-point of the discrete WaveHoltz iteration converges to the discrete
Helmholtz solution with the order of the time-stepper chosen. We present
numerical examples and demonstrate that it is possible to completely remove
time discretization error from the WaveHoltz solution through careful analysis
of the discrete iteration together with updated quadrature formulas.
