Evolution of linear wave dark matter perturbations in the
  radiation-dominant era
  Linear perturbations of the wave dark matter, or $\psi$ dark matter
($\psi$DM), of particle mass $\sim 10^{-22}$eV in the radiation-dominant era
are analyzed, and the matter power spectrum at the photon-matter equality is
obtained. We identify four phases of evolution for $\psi$DM perturbations,
where the dynamics can be vastly different from the counterparts of cold dark
matter (CDM). While in late stages after mass oscillation long-wave $\psi$DM
perturbations are almost identical to CDM perturbations, some subtle
differences remain, let alone intermediate-to-short waves that bear no
resemblance with those of CDM throughout the whole evolutionary history. The
dissimilarity is due to quantum mechanical effects which lead to severe mode
suppression. We also discuss the axion model with a cosine field potential. The
power spectrum of axion models are generally almost identical to those of
$\psi$DM, but in the extreme case when the initial axion angle is near the
field potential top, this axion model predict a power excess over a range of
wave number and a higher spectral cutoff than $\psi$DM as if $\psi$DM had a
higher particle mass.
