# Revisiting terrain-following Boussinesq equations on a highly variable periodic bed

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## Abstract

On re-deriving the Boussinesq-type equations in a terrain-following coordinate system based on the conformal map for an arbitrarily (submerged) periodic bed, it is pointed out that some aspects of the existing approach need to be clarified, specifically the use of velocity potential gradient in the mapped plane in connection with the transformation of velocity vector, and the appropriate frequency–wavenumber relation that is from the solution to the linearized Boussinesq equations. It is shown that as the bed undulation height increases, this relationship increasingly departs from the flat-bottom dispersion relation that has previously been assumed for the terrain-following Boussinesq systems. Over a highly variable periodic bed, the waveforms of linear time harmonic waves already have features reminiscent of nonlinear waveforms, which should be distinguished from the subsequent nonlinear evolution.

## Keywords

Boussinesq equations Terrain-following Conformal mapping## Notes

### Acknowledgements

This research was funded by the Office of Naval Research under program element 0602235N. The author thanks anonymous reviewers for their helpful comments.

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