Speed of Medium Compared to Wave Speed

Do the particles in a medium oscillate about their equilibrium position at the same speed as the wave moves through the medium? Is it the same for both transverse and longitudinal waves?

Suppose a wave $y(x,t)=A\sin\left (2\pi f t - \frac{2\pi x}{\lambda}\right )$ where $y$ is the displacement of a particle from its equilibrium position which is a distance $x$ from an origin at a time $t$.
$f$ is the frequency of the wave and $\lambda$ is the wavelength.
The particle moves with simple harmonic motion, amplitude $A$ and frequency $f$ so the maximum speed of the particle is $2\pi f A$ and its minimum speed is zero.
The wave moves at a constant speed of $f\lambda$.