Linear and nonlinear refraction and Bragg scattering of water waves

The mild slope equation (MSE) is widely used in water wave refraction-diffr action studies. An augmented extension of MSE is derived using operational calculus. It accounts for all the terms that are linear in the derivatives (to any order) of the depth. Consequently, partial previous extensions of M SE are rigorously derived. They are accurate, but only at exact Bragg reson ance. This explains their success in scattering problems. Then, a nonlinear augmented MSE is derived. It extends the accuracy and the range of validit y of existing models for nonlinear shoaling and scattering. [S1063-651X(99) 51702-9].