  We investigate the effects of wave localization on the delay time tau
(frequency sensitivity of the scattering phase shift) of a wave transmitted
through a disordered wave guide. Localization results in a separation
tau=chi+chi' of the delay time into two independent but equivalent
contributions, associated to the left and right end of the wave guide. For N=1
propagating modes, chi and chi' are identical to half the reflection delay time
of each end of the wave guide. In this case the distribution function P(tau) in
an ensemble of random disorder can be obtained analytically. For $N>1$
propagating modes the distribution function can be approximated by a simple
heuristic modification of the single-channel problem. We find a strong
correlation between channels with long reflection delay times and the dominant
transmission channel.
