A theoretical study is made of the free‐surface flow induced by a wavemaker, performing torsional oscillations about a vertical axis, in a shallow rectangular channel near a cut‐off frequency. Exactly at cut‐off, linearized water‐wave theory predicts a temporally unbounded response due to a resonance phenomenon. It is shown, through a perturbation analysis using characteristic variables, that the nonlinear response is governed by a forced Kadomtsev—Petviashvili (KP) equation with periodic boundary conditions across the channel. This nonlinear initial‐boundary‐value problem is investigated analytically and numerically. When surface‐tension effects are negligible, the nonlinear response reaches a steady state and exhibits jump phenomena. On the other hand, in the high‐surface‐tension regime, no steady state is obtained. These results are discussed in connection with similar forced wave phenomena studied previously in a deepwater channel and related lab