# Near-critical reflection of internal waves

## TL;DR

This paper investigates the reflection behavior of internal waves at critical angles on sloping boundaries, demonstrating that solutions can be accurately approximated by incident, reflected, and boundary layer components.

## Contribution

It provides a rigorous mathematical proof that solutions exist and are well approximated in critical reflection geometry, confirming prior predictions.

## Key findings

- Solution exists and is well approximated by wave components
- Reflected second harmonic and boundary layers are significant
- Validates previous theoretical predictions

## Abstract

Internal waves describe the (linear) response of an incompressible stably stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes' laws, and it is expected to be singular if the slope has the same inclination as the group velocity. In this paper, we prove that in this critical geometry the weakly viscous and weakly nonlinear wave equations have actually a solution which is well approximated by the sum of the incident wave packet, a reflected second harmonic and some boundary layer terms. This result confirms the prediction by Dauxois and Young, and provides precise estimates on the time of validity of this approximation.