Long waves in a uniform channel of arbitrary cross-section

Equations of motion are derived for long gravity waves in a straight uniform channel. The cross-section of the channel may be of any shape provided that it does not have gently sloping banks and it is not very wide compared with its depth. The equations may be reduced to those for two-dimensional motion such as occurs in a rectangular channel. The order of approximation in these equations is sufficient to give the solitary wave as a solution.