  We describe radio observations at 244 and 610 MHz of a sample of 20 luminous
and ultra-luminous IRAS galaxies. These are a sub-set of a sample of 31 objects
that have well-measured radio spectra up to at least 23 GHz. The radio spectra
of these objects below 1.4 GHz show a great variety of forms and are rarely a
simple power-law extrapolation of the synchrotron spectra at higher
frequencies. Most objects of this class have spectral turn-overs or bends in
their radio spectra. We interpret these spectra in terms of free-free
absorption in the starburst environment.
  Several objects show radio spectra with two components having free-free
turn-overs at different frequencies (including Arp 220 and Arp 299), indicating
that synchrotron emission originates from regions with very different emission
measures. In these sources, using a simple model for the supernova rate, we
estimate the time for which synchrotron emission is subject to strong free-free
absorption by ionized gas, and compare this to expected HII region lifetimes.
We find that the ionized gas lifetimes are an order of magnitude larger than
plausible lifetimes for individual HII regions. We discuss the implications of
this result and argue that those sources which have a significant radio
component with strong free-free absorption are those in which the star
formation rate is still increasing with time.
  We note that if ionization losses modify the intrinsic synchrotron spectrum
so that it steepens toward higher frequencies, the often observed deficit in
fluxes higher than ~10 GHz would be much reduced.
