On the Korteweg—de Vries equation for a gradually varying channel

摘要: Two integral invariants of Shuto's (1974) generalization the Korteweg—de Vries equation for a unidirectional wave in channel gradually varying breadth b and depth d are derived. The second-order (in amplitude) invariant measures energy, as expected, but first-order mass divided by b½d¼; accordingly, is conserved only if either mean free-surface displacement vanishes or bd½ constant. This difficulty associated with reflected that excited variation neglected KdV approximation. total flux resolved into primary (KdV) residual proportional to wave. constructed neglecting both nonlinearity dispersion (even though significant wave). results applied slowly cnoidal wave, which fully determined conservation energy known uniform channel, solitary not trailing residuals excited. development Boussinesq equations their reduction sketched an appendix.