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(') of the delay time into two independent but equivalent contributions, associated to the left and right end of the waveguide. For N=1 propagating modes, chi and chi(') are identical to half the reflection delay time of each end of the waveguide. 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.