Step 1: Understanding the Concept:
For a traveling wave $y = A \sin(\omega t - kx)$, the wave velocity is how fast the shape moves through space. Particle velocity is how fast a specific part of the medium oscillates up and down.
Step 2: Key Formula or Approach:
1. Wave Velocity: $v = \frac{\omega}{k}$
2. Max Particle Velocity: $V_{p,max} = A\omega$
3. Ratio: $A \times k$ (unitless if units consistent)
Step 3: Detailed Explanation:
From the equation: $A = 60 \mu\text{m} = 60 \times 10^{-6}\text{ m}$, $\omega = 1200 \text{ rad/s}$, $k = 6 \text{ m}^{-1}$.
1. Wave velocity $v = \frac{1200}{6} = 200 \text{ m/s}$.
2. Max particle velocity $V_{p,max} = A\omega = (60 \times 10^{-6}) \times 1200 = 7.2 \times 10^{-2} \text{ m/s}$.
3. Ratio Calculation:
\[ \text{Ratio} = \frac{V_{p,max}}{v} = \frac{7.2 \times 10^{-2}}{200} = \frac{7.2 \times 10^{-2}}{2 \times 10^2} = 3.6 \times 10^{-4} \]
Step 4: Final Answer:
The ratio is $3.6 \times 10^{-4}$.