Answer
The ratio does not depend on the material.
Work Step by Step
We can find an expression for the maximum particle speed:
$v_{max} = y_m~\omega$
We can find an expression for the wave speed:
$v = \frac{\omega}{k} = \frac{\omega}{2\pi/\lambda} = \frac{\omega~\lambda}{2\pi}$
We can find the ratio of $\frac{v_{max}}{v}$:
$\frac{v_{max}}{v} = \frac{y_m~\omega}{\frac{\omega~\lambda}{2\pi}}$
$\frac{v_{max}}{v} = \frac{2\pi ~y_m}{\lambda}$
The ratio is $~~\frac{2\pi ~y_m}{\lambda}$
We can see that this ratio only depends on the amplitude and the wavelength. The ratio does not depend on the linear density of the material.
Therefore, the ratio does not depend on the material.
