Undular Bore Equation

The plane problem of the motion of a wave front over the surface of an ideal incompressible fluid of finite depth at a constant velocity is considered. The initial solution in the form of a smooth bore tends to a steady-state flow. A time-dependent solution in the form of a second-order nonlinear equation is obtained. The stationary form of the equation is compared with well-known results by Lavrent’ev and Korteweg-de Vries (KdV). The linearization agrees with the Airy theory with high accuracy. The accuracy of the solution is estimated numerically. The result, in the form of a nonstationary wave bore, agrees with observations.