A bidirectional long-wave model

For modeling weakly nonlinear and weakly dispersive long gravity waves of typical amplitude a and typical length A, propagating in both directions in a straight, gradually varying channel of breath b(x) and mean water depth h(x), the Boussinesq equations provide a versatile model, with its validity based on the assumptions that a = a/h where.C+ an d C-denote, respectively, right-going and left-going waves, both of 0(e), and Ci is a term of 0( 2 ) representing the interaction between + and C- The evolution equations obtained for + and _ exhibit extensions of the Korteweg-de Vries equation to comprise the additional effects of slow variations in the admittance, (fc/i 1 / 2 ), of varying channels on evolving waves. Main features of this model include: (i) For wave-and-channel-wall interactions, this model accounts for reflection and transmission of long waves in varying channels while maintaining both mass and energy conserved adiabatically. (ii) For wave-wave interactions, head-on collisions between right-and left-going solitary waves are shown to gain a total phase shift which is an algebraic function of the amplitudes of the colliding waves, (iii) For forced generation of nonlinear waves in varying channels, this model admits weakly resonant disturbances with both right-going and left-going components.