Numerical solutions for waves in linear anisotropic media Holger Waubke Acoustics Research Institute Austrian Academy of Sciences holger.waubke@oeaw.ac.at Abstract The Green function for a monopole test function in an isotropic medium is described by a circular equation in wave- number domain. An extension to an elliptic form is derived using a rescaling of the isotropic Green function. Dipole solutions are simple derivatives of the monopole solution in the isotropic case. In the generalized case fractional derivatives will appear. The elliptic function will be investigated. The formulation gives the freedom to use four independent material parameters. These parameters are only linkable for the isotropic case. Therefore a scaling of solutions for the isotropic case is not possible for any kind of anisotropy. With no theoretical limitation any anisotropic linear material can be investigated using the Fourier integral transform of the horizontal directions and solving the vertical direction by analytical solution. The basic analytical solution is determined using the eigenvectors and eigenvalues of the matrix in the completely transformed wave number domain. This solution is simply applied to horizontally stratified media. The back transform has to be done numerically using the fast Fourier transform. Measures to improve the stability of the method are discussed based on an example with critical parameters. 1. Introduction Boundary element methods for orthotropic media are limited to two dimensional cases. This limitation has to do with the definition of orthotropic material that leads to elliptic equations in two dimensions but not for any set of parameters to an elliptic equation in three dimensions. Either a different type of anisotropy that belongs to an elliptic equation in three dimensions or a different numerical method that is capable of dealing with all types of anisotropy is needed. 2. Fourier Integral Transform All applications presented are based on the Fourier integral transform. Using orthogonal coordinates x, y, z in space and time t. Fourier integral transform is used four times about three coordinates in space and time transforming the partial differential equations into simple equations. The basic relations in the time domain are presented in the following subsections. 2.1. Strain Displacement Relationship The strains ε depend directly on the displacements u x,y,z that will be used instead of the velocity as basic variables. This relation includes the compatibility. ( ) ( ) ( ) ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = = z u x u y u z u x u y u z u y u x u x z z y y x z y x zx yz xy zz yy xx 2 1 2 1 2 1 ε ε ε ε ε ε ε (1) Using the Fourier integral transform gives the spectral relation depending on the wave numbers k x,y,z and angular frequency Ω. u D ε ˆ ˆ , ˆ ˆ ˆ 2 0 2 2 2 0 0 2 2 0 0 0 0 0 0 ˆ ˆ ˆ ˆ ˆ ˆ = = z y x x z y z x y z y x zx yz xy zz yy xx u u u jk jk jk jk jk jk jk jk jk ε ε ε ε ε ε (2) 2.2. Stress Strain Relationship The material behavior defines the stress strain relationship. For any kind of anisotropic linear behavior a flexibility matrix can be defined, that will be inverted to an elasticity matrix. For orthotropic material the matrix is limited to three Young’s modules E, shear modules G and Poisson ratios ν. − − − − − − = xy zx zx z z zx z zx z zx x x xy z zx x xy x G G G E E E E E E E E E 2 1 0 0 0 0 0 0 2 1 0 0 0 0 0 0 2 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 ν ν ν ν ν ν F (3) Mo4.F.4 I - 367