Linear elasticity: Difference between revisions

Revision as of 02:50, 27 February 2007

Revision as of 19:44, 12 April 2007

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where

<math>\kappa</math> is [[Bulk modulus|incompressibility]], and

<math>\mu</math> is [[Shear modulus|rigidity]]. These, two parameters[[Elastic aremoduli|modulus alsoof calledelasticity]]. If the [[Lamématerial parameters]]is homogeneous (i.e. the elasticity tensor is constant throughout the material), the acoustic operator becomes:

If the material is homogeneous (i.e. the elasticity tensor is constant throughout the material), the acoustic operator becomes:

:<math>A_{ij}[\nabla]=\alpha^2 \partial_i\partial_j+\beta^2(\partial_m\partial_m\delta_{ij}-\partial_i\partial_j)\,</math>

and the acoustic algebraic operator becomes

:<math>A_{ij}[\mathbf{k}]=\alpha^2 k_ik_j+\beta^2(k_mk_m\delta_{ij}-k_ik_j)\,</math>

where

:<math> \alpha^2=\left(\kappa+\frac{4}{3}\mu\right)/\rho \qquad \beta^2=\mu/\rho </math>

are the [[eigenvalue]]s of <math>A[\hat{\mathbf{k}}]</math> with [[eigenvector]]s <math>\hat{\mathbf{u}}</math> parallel and orthogonal to the propagation direction <math>\hat{\mathbf{k}}</math>, respectively. In the seismological literature, the corresponding plane waves are called P-waves and S-waves (see [[Seismic wave]]).