Partial differential equations with time delays serve as models for systems where both spatial structure and memory effects are important. The asymptotic stability of travelling waves in such systems is still determined entirely by the spectrum of the linearization about the wave. We compare the spectra of localized waves on the unbounded real line with spectra computed on large intervals with appropriate boundary conditions applied at their end points. We show that the spectrum on large intervals approximates the spectrum on the real line when periodic boundary conditions are used. If separated boundary conditions are applied, it is the so-called absolute spectrum together with the extended point spectrum that is approximated; their union typically differs from the spectrum on the real line.