Advertisement

Archive for Rational Mechanics and Analysis

, Volume 216, Issue 3, pp 881–903 | Cite as

Green’s Functions of Wave Equations in \(\mathbb{R}^n_+\times \mathbb{R}_+\)

  • Shijin Deng
  • Weike Wang
  • Shih-Hsien YuEmail author
Article

Abstract

We study the d’Alembert equation with a boundary. We introduce the notions of Rayleigh surface wave operators, delayed/advanced mirror images, wave recombinations, and wave cancellations. This allows us to obtain the complete and simple formula of the Green’s functions for the wave equation with the presence of various boundary conditions. We are able to determine whether a Rayleigh surface wave is active or virtual, and study the lacunas of the wave equation in three dimensional with the presence of a boundary in the case of a virtual Rayleigh surface wave.

Keywords

Wave Equation Surface Wave Fundamental Solution Rayleigh Wave Neumann Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah M.F., B ott R., Gårding: Lacunas for hyperbolic differential operators with constant coefficients.. I. Acta Math 124, 109–189 (1970)CrossRefzbMATHGoogle Scholar
  2. 2.
    Hörmander L.: The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients. Springer, Berlin (2004)Google Scholar
  3. 3.
    Hörmander L.: The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Springer, Berlin (2007)Google Scholar
  4. 4.
    John F.: Partial Differential Equations, 4th edn. Springer, Berlin (1981)Google Scholar
  5. 5.
    Keller J., B lank A.: Diffraction and reflection of pulses by wedges and corners. Commun. Pure Appl. Math. 4, 75–94 (1951)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Keller J.B.: Geometric theory of diffraction. J. Opt. Soc. Am. 52(2), 116–130 (1962)CrossRefADSGoogle Scholar
  7. 7.
    Keller J.B.: Oblique derivative boundary conditions and the image method. SIAM J. Appl. Math. 41(2), 294–300 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kreiss H.-O.: Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23, 277–298 (1970)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Liu T.-P., Y u S.-H.: On boundary relation for some dissipative systems. Bull. Inst. Math. Acad. Sin. (NS) 6(3), 245–267 (2011)zbMATHGoogle Scholar
  10. 10.
    Liu T.-P., Y u S.-H.: Dirichlet–Neumann kernel for hyperbolic-dissipative system in half-space. Bull. Inst. Math. Acad. Sin. (N.S.) 7(4), 477–543 (2012)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Liu, T.-P., Y u, S.-H.: Wave propagator and determinant of Dirichlet–Neumann kernel map, preprintGoogle Scholar
  12. 12.
    Petrovsky I.G.: On the diffusion of waves and the lacunas for hyperbolic equations (in English). Recueil Mathmatique (Matematicheskii Sbornik) 17(59), 289–368 (1945)Google Scholar
  13. 13.
    Polyanin, A.D.: Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman & Hall/CRC, Boca Raton, (2002)Google Scholar
  14. 14.
    Rayleigh, L.: On waves propagated along the plane surface of an elastic solid. Proc. Lond. Math. Soc. s1-17, 4–11 (1885)Google Scholar
  15. 15.
    Sakamoto R.: E-well posedness for hyperbolic mixed problems with constant coefficients. J. Math. Kyoto Univ. 14, 93–118 (1974)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Sakamoto, R.: Hyperbolic Boundary Value Problems. Cambridge University Press, Cambridge, 1982 [Translated from the Japanese by Katsumi Miyahara]Google Scholar
  17. 17.
    Serre D.: Solvability of hyperbolic IBVPs through filtering. Methods Appl. Anal. 12(3), 253–266 (2005)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Serre D.: Second order initial boundary-value problems of variational type. J. Funct. Anal. 236(2), 409–446 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Wang, H., Y u, S.-H., Z hang, X.: On Rayleigh wave and Lamb’s Problem, preprintGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Department of MathematicsNational University of SingaporeSingaporeSingapore

Personalised recommendations