Answer
The ratio is $~~\frac{2\pi ~y_m}{\lambda}$
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 $~~\frac{2\pi ~y_m}{\lambda}$
