Anisotropic Gaussian wave models

Abstract

Let $d$ be an integer greater or equal to 2 and let $\mathbf k$ be a $d$-dimensional random vector. We call Gaussian wave model with random wavevector $\mathbf k$ any stationary Gaussian random field defined on $\mathbb{R}^d$ with covariance function $t\mapsto \mathbb{E}[\cos(\mathbf k.t)]$. Any stationary Gaussian random field on $\mathbb{R}^d$ can be studied as a random wave. The purpose of the present paper is to link properties of the random wave with the distribution of the random wavevector, with a focus on geometric properties. We mainly concentrate on random waves such that the distribution of the norm of the wavevector and the one of its direction are independent. In the planar case, we prove that the expected length of the nodal lines is decreasing as the anisotropy of the wavevector is increasing, and we study the direction that maximizes the expected length of the crest lines. We illustrate our results on two specific models: a generalization of Berry's monochromatic planar waves and a spatiotemporal sea wave model whose random wavevector is supported by the Airy surface in $\mathbb{R}^3$. According to a general theorem, these two Gaussian fields are anisotropic almost sure solutions of partial differential equations that involve the Laplacian operator: $\Delta f+\kappa^2f=0$ (where $\kappa=\|\mathbf k\|$) for the former, $\Delta f+\partial^4_tf=0$ for the latter.