Notes on diffusive and shear quasinormal modes of black branes

In the literature, to extract the dispersion relation of low-frequency quasinormal modes in both diffusive and shear channels, it is a customary recipe to assume firstly $\omega \sim \mathcal{O}$ to solve the equation of motion and finally $\omega \sim \mathcal{O}(q^2)$ when applying the Dirichlet boundary condition. The two assumptions appear confusing though the recipe usually gives the same result as that from other channels or from the Kubo formula. We refine the recipe by assuming $\omega \sim \mathcal{O}(q^2)$ from the beginning to the end, and demonstrate it in the diffusive channel of the Schwarzschild black brane and the shear channel of the Gauss-Bonnet black brane.