Linear and nonlinear refraction and Bragg scattering of water waves

The mild slope equation (MSE) is widely used in water wave refraction-diffraction 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 MSE are rigorously derived. They are accurate, but only at exact Bragg resonance. This explains their success in scattering problems. Then, a nonlinear augmented MSE is derived. It extends the accuracy and the range of validity of existing models for nonlinear shoaling and scattering.