From ray tracing to topological waves in continuous media

Abstract

Inhomogeneous media commonly support a discrete number of wave modes that are trapped along interfaces defined by spatially varying parameters. In the case of multicomponent wave problems, those trapped modes fill a frequency gap between different wave bands. When they are robust against continuous deformations of parameters, such waves are said to be topological. It has been realized over the last decades that the existence of such topological waves can be predicted by computing a single topological invariant, the first Chern number, in a dual bulk wave problem that is much simpler to solve than the original wave equation involving spatially varying coefficients. The correspondence between the simple bulk problem and the more complicated interface problem is usually justified by invoking an abstract index theorem. Here, by applying ray tracing machinery to the paradigmatic example of equatorial shallow water waves, we propose a physical interpretation of this correspondence. We first compute ray trajectories in a phase space given by position and wavenumber of the wave packet. We then apply a quantization condition to describe the spectral properties of the original wave operator. This bridges the gap between previous work by Littlejohn and Flynn showing manifestation of Berry curvature in ray tracing equations, and more recent studies that computed the Chern number of flow models by integrating the Berry curvature over a closed surface in parameter space. We find that an integral of Berry curvature over this closed surface emerges naturally from the quantization condition, which allows us to recover the bulkinterface correspondence. Contents I. Motivation from equatorial shallow water waves A. Wave equation and definition of the spectral flow B. The semi-classical limit for equatorial shallow water waves C. Strategy to interpret the spectral flow II. Scalar wave operator and symbols A. From multicomponent to scalar wave operator B. Expression of the scalar operator in the semi-classical limit C. Asymptotic expansion up to order one D. Application to the shallow water wave problem III. wave packet dynamics in the semi-classical regime A. Wave packet center of mass and wave packet momentum B. WKB ansatz for the scalar wavefield C. Assumption of a localized wave packet D. Ray tracing equations 1. Gauge dependent, canonical Hamiltonian form 2. Gauge independent, non-canonical Hamiltonian form 3. Application to shallow water waves IV. Topological properties of Matsuno symbol 1. Matsuno symbol and Kelvin plane waves. 2. Berry curvature for eigenvectors of Matsuno symbol