The study of the propagation of undular bores in channels is relevant to many applications which go from the propagation of tsunami waves, to that of tidal bores/waves, to the propagation of strong waves in manmade channels due to hazards (e.g. dam breaking). In absence of banks, the flow exhibits a transition across which undulating waves  start breaking and transform in fully breaking turbulent bores. In presence of banks, a double transition is observed in which the flow goes first from a quasi-1D topology to a fully 3D topology, and then to a fully breaking turbulent bore. The transition is in any case governed the Froude number. The wavelengths associated to the first two regimes are very different, the low Froude undulating waves (first kind) have a much longer wave length. While the shorter waves (second kind) can be explained and analyzed by means of classical wave dispersion, no mechanism is known to explain the longer low Froude waves.

For all the applications mentioned, a  trapezoidal section approximation is an excellent one, relevant for flows both natural and manmade environments. This has led to many experimental campaigns in this configuration, the most well known being those due to Treske (1994), for which a full set of data is available.

In this work, we investigate the first transition across which a dramatic reduction of the wavelength is observed, and which is absent in straight channels.  We propose  a new, fully hydrostatic, dispersive like mechanism for the appearance of these waves. This mechanism is related to wave refraction in the traversal direction, and is a purely geometric dispersive effect. An asymptotic analysis is developed to verify this hypothesis, to propose a dispersion relation which is used to  predict the wavelength-Froude dependency using the theory proposed by Lemoine (1948).