  We revisit the analysis by R.A. Gardner of convergence of spectra of periodic
traveling waves in the homoclinic, or infinite-period limit, extending his
results to the case of essential rather than point spectra of the limiting
homoclinic wave. Notably, convergence to essential spectra is seen to be of
algebraic rate with respect to period as compared to the exponential rate of
convergence to point spectra. In the course of the analysis, we show not only
convergence of spectrum but also convergence of an appropriate renormalization
of the associated periodic Evans function to the Evans function for the
limiting homoclinic wave, a fact that is useful for numerical investigations
