For some time the magnetotail has been considered as a possible region where hydromagnetic waves can propagate as waveguide modes. Recently, attention has turned to the magnetospheric flanks as waveguides, and much useful insight has been gained into propagation of fast waveguide modes there, and the structure of the field line resonances they can drive. We return to the magnetotail and investigate hydromagnetic wave propagation and coupling in a magnetotail waveguide. This problem is significantly different from the flank waveguide as the ambient magnetic field is directed along the waveguide rather than across. Field line resonances of the flank type are not possible in the lobe waveguide. We describe a numerical simulation of a model waveguide in which the Alfv¿n speed decreases across the waveguide to the central plasma sheet. The waveguide is stimulated by a short compressional perturbation located in the far tail. The cross-tail spatial structure is chosen to give relatively weak coupling between fast and Alfv¿n modes so that phase and group velocities of uncoupled fast modes can be used to interpret the results. We find that the perturbation propagates dispersively down the waveguide in the form of fast waveguide modes. Fourier components with small parallel wavenumber contain most of the energy, and propagate relatively slowly toward the Earth. These act as moving sources which launch Alfv¿n waves continuously earthward. The wave dispersion relations are such that the waveguide modes couple with Alfv¿n waves only in a limited region of the transverse Alfv¿n speed gradient. The Alfv¿n waves travel at the local Alfv¿n speed along each field line, so that as they travel the wave on a given field line becomes increasingly out of phase with waves on adjacent field lines. The phase mixing in our model is novel in that it includes the effects of transverse gradients in both Alfv¿n frequency and parallel wavenumber which tend to cancel each other out. Nevertheless, the phase-mixing process leads to increasingly fine transverse structure as the waves progress down the waveguide. The results are likely to be applicable in regions such as the plasma sheet boundary layer and the plasma mantle. ¿ 1998 American Geophysical Union