An Asymptotic Average Decay Rate for the Wave Equation with Variable Coefficient Viscous Damping


The analysis of the damping rate for eigenmodes of a vibrating system with variable coefficient viscous damping is usually difficult because no explicit formulas are available in general. In this paper, using the one-dimensional wave equation as a model, it is shown that for such a system there is an asymptotic average decay rate of eigenmodes that is equal to the uniform damping rate of high frequencies of a wave equation with homogenized constant damping coefficient. The proof is obtained by asymptotic estimation of eigenfrequencies based on an earlier work of Birkhoff and Langer [Proc. Amer. Acad. Arts Sci., 58 (1923), pp. 51–128]. Numerical confirmation is also included, with details of computations given in [C. Qi, Computing the Spectrum of a Vibrating String with Positive and Negative Locally Distributed Viscous Damping, Masters Thesis, Pennsylvania State University, 1987].

MSC codes

  1. 35P20


  1. viscous damping
  2. eigenfrequencies
  3. vibration

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G. D. Birkhoff, R. E. Langer, The boundary problems and developments associated with a system of ordinary linear differential equations of the first order, Proc. Amer. Acad. Arts Sci., 58 (1923), 51–128
Goong Chen, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control Optim., 17 (1979), 66–81
C. Qi, Masters Thesis, Computing the spectrum of a vibrating string with positive and negative locally distributed viscous damping, Masters Thesis, Department of Mathematics, Pennsylvania State University, University Park, PA, 1987, August

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Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1341 - 1347
ISSN (online): 1095-712X


Submitted: 24 August 1988
Accepted: 5 July 1989
Published online: 10 July 2006

MSC codes

  1. 35P20


  1. viscous damping
  2. eigenfrequencies
  3. vibration



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