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The characteristics of topologically protected wave propagation is typicallypredicted via the band structure of the primitive unit cell, using Berrycurvature to predict localized interface or boundary states (as well as theirdegree of localization), and the dispersion relation to predict propagatinggroup velocity. However, practical systems are finite in size and are driven byexcitation with a finite bandwidth as guaranteed by the Fourier uncertaintyprinciple. Hence, the dynamics predicted by the ideal infinite systems drivenat a given frequency deviate from practical topological systems driven across afinite bandwidth. In this work, we demonstrate that the propagating topologicalwaves in a valley Hall system can be interpreted using the underlying lineardegenerate modal basis. We show that only a small subset of closely spacedmodes with the appropriate phase differences comprise the topological waves,which enables the construction of analytical reduced-order models thataccurately capture the topological wave. This result is related to the sparsedensity of the modal spectrum inside the topological band gap, which allows forthe energy to be carried nearly completely by a narrow band of spectrallyisolated eigenmodes that are spatially localized along the domain boundary.This finding is subsequently employed to predict the true group velocity ofpropagating topological waves with respect to input signal bandwidth andfrequency by matching each active mode to corresponding band locations of thesupercell dispersion diagram describing the semi-infinite system. Moreover, wedemonstrate that damped topological waves may be studied within the frameworkof modal superposition by utilizing the damped modal spectrum to characterizethe variation topological wave group velocity and predict edge-to-bulktransitions.

A Modal Decomposition Approach to Topological Wave Propagation