sound waves, and magneto-elastic (me) waves, a combination of Alfvén and shear waves. The complete eigenvalue structure of relativistic ideal MHD equations in the presence of an elastic solid is not known. To make a simple order of magnitude estimate of the different wave speeds, we use the expression of magneto-elastic torsional waves parallel to the magnetic field derived in [@Gabler2012] as well as the expression for fast magnetosonic waves perpendicular to the field[^3]: $$\begin{aligned}
v_{\rm me} /c = \sqrt{\frac{\mu_s+B^2}{e + B^2}} \qquad \qquad v_{\rm ms} /c= \sqrt{\frac{e c_s^2 + B^2}{e+B^2}},\end{aligned}$$ where $e$ is the energy density and $\mu_s$ the shear modulus. Note that in the limit of low magnetic field ($B^2\ll \mu_s, \, B^2\ll e$) we recover the shear and sound speed, respectively. In the high magnetic field limit ($B^2\gg \mu_s,\, B^2\gg e$) both, $v_{\rm me}$ and $v_{\rm ms}$, coincide with the speed of light. Inside the fluid core ($\mu_s=0$) the magneto-elastic speed becomes the Alfvén speed.

Figure \[fig:speeds\] shows the value of the characteristic speeds in the outer layers of a typical NS model for different magnetic fields in the magnetar range. Indeed, both fast magnetosonic waves and Alfvén waves have a degenerate speed equal to the speed 