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I'm looking at the vibration of a solid having a lattice structure, they obey the following equation: $$\rho\partial_t^2u_i = C_{ijkl}\nabla_j\nabla_ku_{l}$$ with $u(\vec{x},t)$ the displacement to the equilibrium position, $C_{ijkl}$ the stiffness matrix, $\rho$ the mass density. Knowing the following shape for the stiffness (in the Voigt notation):$$\begin{pmatrix}a&b&b&0&0&0\\b&a&b&0&0&0\\b&b&a&0&0&0\\0&0&0&c&0&0\\0&0&0&0&c&0\\0&0&0&0&0&c\end{pmatrix}$$ I do end with the following differential equation: $$\begin{pmatrix}\rho\partial_t^2-a\partial_x^2& (b+c)\partial_x\partial_y&(b+c)\partial_x\partial_z\\ (b+c)\partial_x\partial_y&\rho\partial_t^2-a\partial_y^2&(b+c)\partial_y\partial_z\\ (b+c)\partial_x\partial_z&(b+c)\partial_y\partial_z&\rho\partial_t^2-a\partial_z^2&\end{pmatrix}\begin{pmatrix}u_x\\u_y\\u_z\end{pmatrix}=0$$ And I would like to derive a relation dispersion from there but I am not sure of the following steps, should go to Fourrier space and look for calculate the values of $\omega$ that would make the determinant vanish, or diagonalize it first?

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From what I can remember in similar problems, in order to

$$\begin{pmatrix}\rho\partial_t^2-a\partial_x^2& (b+c)\partial_x\partial_y&(b+c)\partial_x\partial_z\\ (b+c)\partial_x\partial_y&\rho\partial_t^2-a\partial_y^2&(b+c)\partial_y\partial_z\\ (b+c)\partial_x\partial_z&(b+c)\partial_y\partial_z&\rho\partial_t^2-a\partial_z^2&\end{pmatrix}\begin{pmatrix}u_x\\u_y\\u_z\end{pmatrix}=0$$

has a solution, is needed that,

$$\det\begin{pmatrix}\rho\partial_t^2-a\partial_x^2& (b+c)\partial_x\partial_y&(b+c)\partial_x\partial_z\\ (b+c)\partial_x\partial_y&\rho\partial_t^2-a\partial_y^2&(b+c)\partial_y\partial_z\\ (b+c)\partial_x\partial_z&(b+c)\partial_y\partial_z&\rho\partial_t^2-a\partial_z^2&\end{pmatrix} = 0$$,

so you need to solve this, and it gives you a multiple (I guess four) order equation that represents the dispersion relation (with all it's multiple branches.)

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