A wave on a stretched rope is y = 0.025 sin(8x - 120t), with x and y in metres and t in seconds…

Step 1. Wave speed. $v = \dfrac{120\ \text{rad/s}}{8\ \text{rad/m}} = 15\ \text{m/s}$.

Step 2. The particle. Each particle moves in simple harmonic motion with $\omega = 120\ \text{rad/s}$ and amplitude $0.025\ \text{m}$. Its largest speed is $a\omega = 0.025\ \text{m} \times 120\ \text{rad/s} = 3.0\ \text{m/s}$. Its largest acceleration is $a\omega^2 = 0.025\ \text{m} \times (120\ \text{rad/s})^2 = 360\ \text{m/s}^2$.

Step 3. One moment. The phase is $8\ \text{rad/m} \times 0.30\ \text{m} - 120\ \text{rad/s} \times 0.010\ \text{s} = 2.4\ \text{rad} - 1.2\ \text{rad} = 1.2\ \text{rad}$. So $u = -3.0\ \text{m/s} \times \cos(1.2) = -3.0\ \text{m/s} \times 0.362 = -1.09\ \text{m/s}$. The particle is moving down.

Notice. The wave moves at $15\ \text{m/s}$. No particle ever moves faster than $3\ \text{m/s}$. The two speeds are different things.