On the Role of Surface Tension in the Energy Budget of Dispersive Undular Bores

Undular bores are classical shallow-water phenomena in which a sharp transition between two flow states is replaced, in a dispersive theory, by an oscillatory wave train. In non-dispersive shallow-water theory, the bore is associated with an apparent loss of mechanical energy. In dispersive models, this energy can be interpreted as being redistributed into the oscillatory tail. The aim of this short article is to formulate a possible extension of this interpretation when surface tension is included. The capillary contribution modifies the long-wave dispersion coefficient through a Bond-number-dependent term and adds an additional surface energy to the total energy functional. We derive the basic capillary-gravity KdV scaling, identify the modified energy density, and discuss how surface tension may affect the amplitude, wavelength, and energy flux of the trailing oscillations. The proposed direction is relevant for small-scale laboratory bores, tidal-bore fronts, and shallow tidal currents in which a rapid transition generates short dispersive oscillations. Special attention is paid to the critical value Bo=1/3, where the classical KdV dispersion vanishes, and a fifth-order correction is required.