## Wave Propagation Analysis with Boundary Element MethodTime-dependent problems, that are frequently modelled by hyperbolic partial differential equations, can be dealt with the boundary integral equations (BIEs) method. The ideal situation is when the partial differential equation is homogeneous with constant coefficients, the initial conditions vanish and the data are given only on the boundary of a domain independent of time. In this situation the transformation of the differential problem to a BIE follows the same well-known method for elliptic boundary value problems. In fact the starting point for a BIE method is the representation. of the differential problem solution in terms of single layer and double layer potentials using the fundamental solution of the hyperbolic partial differential operator. |

### Contents

Basic theory | 1 |

The onedimensional wave problem | 15 |

The two dimensional wave problem | 41 |

Discretization | 75 |

A Appendix | 103 |

116 | |

### Common terms and phrases

analysis analytical inner integration approximate solution as)ē bilinear form boundary element method boundary integral equations Cauchy-Schwarz inequality coerciveness property consider constant shape functions continuous and coercive crack defined dimensional region Dirichlet problem discretization displacement double integration domain Duong dzds end-points energetic weak formulation energy Figure formula Inner integral fundamental solution Galerkin Galerkin approximation Gauss-Legendre formula Inner HFP formula 4.34 incident wave inequality instants integrand function kernel Lē(E linear shape functions matrix Neumann boundary conditions Neumann problem nodes numerical evaluation numerical results operator Outer integral Paley-Wiener theorem polynomial product rule 4.30 quadratic form quadrature regularization procedure 4.27 Relative Error representation formula respect rewrite the double sinē singularity space space-time stability Theorem vanishing w(di wave equation wave problem wave propagation weak problem ди дп