Unstable Surface Modes in Finite Chain Computations:Deficiency of Reflection Coefficient Approach

In this paper,we investigate the stability for a finite harmonic lattice under a certain class of boundary conditions.A rigorous eigenvalue study clarifies that the invalidity of Fourier modes as the basis results in the deficiency of standard reflection coefficient approach for stability analysis.In a certain parameter range,unstable surface modes exist in the form of exponential decay in space,and exponential growth in time.An approximate eigen-polynomial is proposed to ease the stability analysis.Moreover,the eigenvalues with small positive real part quantitatively explain the long time instability in wave propagation computations.Numerical results verify the analysis.