Motivated by earlier calculations showing large effects when neutrinos
propagate through fluctuating media, we perform here a detailed analysis of how
density fluctuations in the sun in the form of helioseismic waves can modify
the MSW solution to the solar neutrino problem. We find negligible effects for
the MSW spectrum, even under extreme circumstances. There are two main reasons
why our conclusions differ from earlier analyses. First, most helioseismic
waves do not affect neutrino propagation because their amplitude is too small
in the MSW resonance region, which is the only region to whose fluctuations
neutrinos are sensitive. There is one class of waves which may be subject to an
instability, however, and so can have significantly larger amplitudes. But the
wavelength for these waves is so long that it invalidates the previous methods
of calculation. Our more complete calculation significantly reduces the
prediction for their influence on neutrino propagation.