Weakly dispersive, fully nonlinear waves in channels of arbitrary cross-section are considered from a variational viewpoint. A set of general equations is derived, that resemble the Serre-Green-Naghdi equation. Traveling wave solutions in prismatic channels are derived in the form of pseudo-elliptic integrals. The case of solitary waves in trapezoidal channels is addressed more deeply, with comparison to experiments as regards the celerity of the leading wave in an undular bore.