Write down the Christoffel wave equation for a wave in a cubic crystal propagating along the [111] direction. For the choice C₁₁ = 1, C₁₂ = C₄₄ = 0.5, and ρ = 1, use a program like Mathematica to find the eigenvectors of this matrix, which correspond to the polarizations of the propagating waves, and the velocity of each polarization.