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We study reflectionless properties at the boundary for the wave equation inone space dimension and time, in terms of a well-known matrix that arises froma simple discretisation of space. It is known that all matrix functions of thefamiliar second difference matrix representing the Laplacian in this settingare the sum of a Toeplitz matrix and a Hankel matrix. The solution to the waveequation is one such matrix function. Here, we study the behaviour of thecorresponding waves that we call Toeplitz waves and Hankel waves. We show thatthese waves can be written as certain linear combinations of even Besselfunctions of the first kind. We find exact and explicit formulae for thesewaves. We also show that the Toeplitz and Hankel waves are reflectionless oneven, respectively odd, traversals of the domain. Our analysis naturallysuggests a new method of computer simulation that allows control, so that it ispossible to choose -- in advance -- the number of reflections. An attractiveresult that comes out of our analysis is the appearance of the well-known shiftmatrix, and also other matrices that might be thought of as Hankel versions ofthe shift matrix. By revealing the algebraic structure of the solution in termsof shift matrices, we make it clear how the Toeplitz and Hankel waves areindeed reflectionless at the boundary on even or odd traversals. Although thesubject of the reflectionless boundary condition has a long history, we believethe point of view that we adopt here in terms of matrix functions is new.

The reflectionless properties of Toeplitz waves and Hankel waves: an analysis via Bessel functions