A note on linear dispersion and shoaling properties in extended Boussinesq equations

A set of optimum parameter alpha is obtained to evaluate the linear dispersion and shoaling properties in the extended Boussinesq equations of Madsen and Sorensen (1992), Nwogu (1993), and Chen and Liu (1995). Optimum alpha values are determined to produce minimal errors in each wave property of phase velocity, group velocity, or shoaling coefficient relative to the analytical one given by the Stokes wave theory. Comparisons are made of the percent errors in phase velocity, group velocity, and shoaling coefficient produced by the Boussinesq equations with a different set of optimum alpha values. The case with a fixed value of alpha = -0.4 is also presented in the comparison. The comparisons reveal that the optimum alpha value tuned for a particular wave property gives in general poor results for other properties. Considering all the properties simultaneously, the fixed value of alpha = -0.4 may give overall accuracies in phase velocity and shoaling coefficient for all the types of Boussinesq equations selected in this study. (C) 2003 Elsevier Science Ltd. All rights reserved.