Third-order dispersion in shallow water waves

The effects of the third-order dispersion on the behavior of a wave packet propagating through shallow water in open channels are investigated in this paper. By the Fourier analysis, upon linearization of the governing fully shallow water equations around a uniform state, the complex dispersion relation is obtained. However, it has been found that for wave packet propagation in shallow water flow, our attention can be focused only on the real part of the frequency. Then, for this quantity, the Taylor series approximation about the carrier wave number, including the third-order term, is assumed. For the first, second, and third derivatives of the real part of the frequency with respect to the wave number that represent the dispersion parameters, and for the imaginary part of the group velocity, the effects of the variations of the dimensionless numbers utilized in shallow water wave analyses, such as the Courant number, the Froude number, the friction parameter, and the relative roughness, are examined in the wave number domain. After examining the solution of the fully nonlinear shallow water equations for a simple Gaussian wave, using two different sets of dispersion parameters, the time evolution of a Gaussian wave packet is examined. While analytical expressions in the closed form are available for first- and second-order dispersion approximations, approximate Airy function solutions are adopted when third-order dispersion is taken into account. In this case, the shape of the wave packet, broadened and distorted, acquires an additional oscillatory structure in the form of subwaves of decreasing amplitude as the distance increases. The theoretical results may lead to understanding the role of the third-order dispersion and, at the same time, are useful for high accuracy investigations on numerical solutions of shallow water waves.