## 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 differentialoperator. |

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### 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 bilinear form boundary element method boundary integral equation boundary unknowns Cauchy-Schwarz inequality coerciveness property consider constant shape functions continuous and coercive crack deﬁned density dimensional region Dirichlet problem discretization displacement double integration domain Duong dzds end-points energetic weak formulation ﬁeld Figure ﬁnal ﬁrst ﬁxed formula Inner integral fundamental solution Galerkin Galerkin approximation Gauss-Legendre formula Inner Heaviside function HFP formula 4.34 incident wave inequality instants integrand function kernel linear shape functions matrix mixed boundary conditions Ms,ms Neumann boundary conditions Neumann problem nodes numerical evaluation numerical results operator order centered Outer integral Paley-Wiener theorem parameter polynomial product rule 4.30 quadratic form quadrature regularization procedure 4.27 Relative Error representation formula respect rewrite the double space space-time square root mild stability Theorem vanishing wave equation wave problem wave propagation