Commun. Comput. Phys., 8 (2010), pp. 143-158.

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

Shaoqiang Tang 1*, Ming Fang 1

1 Center for Applied Physics and Technology, and LTCS, College of Engineering, Peking University, Beijing 100871, China.

Received 9 April 2009; Accepted (in revised version) 5 October 2009
Available online 12 February 2010


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.

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PACS: 43.20.+g, 83.10.Rs, 63.20.D-
Key words: Unstable surface mode, reflection coefficient, finite chain.

*Corresponding author.
Email: (S. Tang)

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