Transmission delay times of localized waves

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 waveguide. Localization results in a separation $\tau =\chi +{\chi }^{\prime }$ of the delay time into two independent but equivalent contributions, associated to the left and right end of the waveguide. For $N=1\mathrm{}$ propagating modes, $\chi$ and ${\chi }^{\prime }$ are identical to half the reflection delay time of each end of the waveguide. In this case the distribution function $P\left(\tau \right)$ in an ensemble of random disorder can be obtained analytically. For $N>\mathrm{}1\mathrm{}$ 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.