Improved Korteweg & de Vries type equation with consistent shoaling characteristics

An improved Korteweg & de Vries type equation for uneven water depths with consistent linear shoaling characteristics is derived. The improvement of the equation is with respect to linear dispersion characteristics while consistency in linear shoaling characteristics is achieved via an exact agreement of the shoaling rate of the wave equation with that obtained from the principle of energy flux concept. Improvements in both linear dispersion and linear shoaling properties are demonstrated analytically and numerically by simulating a challenging experimental test case of nonlinear wave propagation over a submerged bar.