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The biharmonic equation |
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denotes propagation direction
and <math>c=\omega/\sqrt{\mathbf{k}\cdot\mathbf{k}}</math> is phase velocity.
== Isotropic homogeneous media ==
In [[Hooke's Law#Isotropic materials|isotropic]] media, the [[elasticity tensor]] has the form
:<math> C_{ijkl}
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<math>\kappa</math> is [[Bulk modulus|incompressibility]], and
<math>\mu</math> is [[Shear modulus|rigidity]].
:<math>A_{ij}[\nabla]=\alpha^2 \partial_i\partial_j+\beta^2(\partial_m\partial_m\delta_{ij}-\partial_i\partial_j)\,</math>
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:<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]]).
== The biharmonic equation ==
From the expression for the acoustic operator for isotropic homogeneous materials, it is seen that:
:<math>\partial_i\partial_jA_{ij}[\nabla]=\alpha^2 \partial_i\partial_i\partial_j\partial_j+\beta^2(\partial_m\partial_m\partial_j\partial_j
-\partial_i\partial_i\partial_j\partial_j)=\alpha^2\partial_i\partial_i\partial_j\partial_j\,</math>
which is just <math>\alpha^2</math> times the [[biharmonic operator]]. It follows that for the elastostatic case, where all variation in time is zero, each of the <math>u_k</math> obey a biharmonic equation:
:<math>\alpha^2\partial_i\partial_i\partial_j\partial_ju_k=-f_k/\rho\,</math>
== References ==
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