Vol. 33

Front:[PDF file] Back:[PDF file]
Latest Volume
All Volumes
All Issues

A Simple Solution for the Damped Wave Equation with a Special Class of Boundary Conditions Using the Laplace Transform

By Namik Yener
Progress In Electromagnetics Research B, Vol. 33, 69-82, 2011


It is proven that for the damped wave equation when the Laplace transforms of boundary value functions ψ(0,t) and (∂ψ(z,t)/∂z)z=0 of the solution ψ(z,t) have no essential singularities and no branch points, the solution can be constructed with relative ease. In such a case while computing the inverse Laplace transform, the integrals along the segments on the real line are shown to always cancel. The integrals along the circles Cε and C'-ε about the point s=-σ/ε determined by the coefficient of the time derivative in the differential equation and point s=0 are shown to vanish unless Laplace transforms of mentioned boundary value functions have poles at these points. If such poles do exist, the problem is nevertheless one of integration along circles about these poles and then setting the radii of these circles equal to zero in the limit.


Namik Yener, "A Simple Solution for the Damped Wave Equation with a Special Class of Boundary Conditions Using the Laplace Transform," Progress In Electromagnetics Research B, Vol. 33, 69-82, 2011.


    1. Lu, X., The applications of microlocal analysis in σ-evolution equations, Ph.D. dissertation, Zhejiang University, Hangzhou, Zhejiang, 2010.

    2. Jradeh, M., "On the damped wave equation," 12th Intern. Confer. on Hyperbolic Problems, Maryland, USA, June 9-13, 2008.

    3. Stratton, J. A., Electromagnetic Theory, McGraw Hill, New York, 1941.

    4. Alexio, R. and E. C. de Oliveira, "Green's functions for the lossy wave equation," Revista Brasileira de Ensino de Fisica, Vol. 30, No. 1, Sao Paulo, 2008.

    5. Zauderer, E., Partial Differential Equations of Applied Mathematics, 2nd Ed., John Wiley & Sons Inc., New York, 1989.

    6. Asfar, O. R., "Riemann-green function solution of transient electromagnetic plane waves in lossy media," IEEE Trans. EM Compatibility, Vol. 32, No. 3, 228-231, 1990.

    7. Lin, H., et al., "A novel unconditionally stable PSTD method based on weighted Laguerrre polynomial expansion," Journal of Electromagnetic Waves and Applications, Vol. 23, No. 8-9, 1011-1020, 2009.

    8. Jung, B. H. and T. K. Sarkar, "Solving time domain Helmholtz wave equation with MOD-FDM," Progress In Electromagnetics Research, Vol. 79, 339-352, 2008.

    9. Levinson, N. and R. M. Redheffer, "Complex Variables," Holden-Day Inc., San Francisco, 1970.

    10. Sonnenchein, E., I. Rutkevich, and D. Censor, "Wave packets and group velocity in absorbing media: Solutions of the telegrapher's equation ," Progress In Electromagnetics Research, Vol. 27, 129-158, 2000.

    11. Churchill, R. V., Modern Operational Mathematics in Engineering, McGraw Hill, New York, 1944.