European Journal of Mechanics B/Fluids 24 (2005) 255–274 A Fourier–Boussinesq method for nonlinear water waves Harry B. Bingham a,∗,1 , Yehuda Agnon b,2 a Mechanical Engineering, Technical University of Denmark, Lyngby, Denmark b Civil Engineering, The Technion, Haifa, Israel Received 3 November 2003; received in revised form 10 May 2004; accepted 20 June 2004 Available online 25 September 2004 Abstract A Boussinesq method is derived that is fully dispersive, in the sense that the error of the approximation is small for all 0  kh < ∞ (k the magnitude of the wave number and h the water depth). This is made possible by introducing the generalized (2D) Hilbert transform, which is evaluated using the fast Fourier transform. Variable depth terms are derived both in mild-slope form, and in augmented mild-slope form including all terms that are linear in derivatives of h. A spectral solution is used to solve for highly nonlinear steady waves using the new equations, showing that the fully dispersive behavior carries over to nonlinear waves. A finite-difference–FFT implementation of the method is also described and applied to more general problems including Bragg resonant reflection from a rippled bottom, waves passing over a submerged bar, and nonlinear shoaling of a spectrum of waves from deep to shallow water.  2004 Elsevier SAS. All rights reserved. Keywords: Boussinesq methods; Nonlinear waves; Coastal and offshore engineering; Bragg reflection 1. Introduction Predicting the nonlinear propagation of dispersive waves over a bathymetry is desirable in many coastal and offshore appli- cations. Realistic problems require analysis over a complicated geometry on the order of a hundred by a hundred significant wavelengths, and in relative water depths all the way from practically infinite to zero. Such problems pose a formidable chal- lenge and are generally treated using potential flow methods. Typically the velocity potential is expanded in a set of basis functions which individually satisfy the Laplace equation; and the expansion coefficients are determined to satisfy the remain- ing conditions on the fluid boundary. The number of degrees of freedom (usually a set of values of the potential or its derivatives on the boundary) is hence significantly smaller than would result from discretizing the entire fluid volume. The basis functions used are typically polynomials, singular Green’s functions, or Fourier functions; each of which has advantages and disadvan- tages depending on the phenomena of primary interest. Boundary integral methods are fully dispersive and simple to apply on complicated geometries, but are relatively computationally demanding since they require discretizing the bottom and lead to full matrix systems. Calculating the Green functions is also more expensive than polynomial or Fourier coefficients. Grilli et al. [1], Wang et al. [2], and Kring et al. [3] are recent examples of the application of boundary element techniques to coastal problems. * Corresponding author. E-mail address: hbb@mek.dtu.dk (H.B. Bingham). 1 Supported by the Danish Research Council (STVF) grant no. 9801635, and the Danish Center for Scientific Computing. 2 Supported by The Fund for the Promotion of Research at the Technion. 0997-7546/$ – see front matter  2004 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechflu.2004.06.006