Step 1: Understanding the Concept:
A progressive wave equation of the form \(y = A \sin(\omega t - kx)\) describes a wave travelling in space over time.
The parameter \(\omega\) is the angular frequency (related to time), and \(k\) is the wave number (related to position).
Wave velocity is the speed at which the wave profile moves.
Key Formula or Approach:
The wave velocity \(v\) is given by the ratio of angular frequency to wave number:
\[ v = \frac{\omega}{k} \]
Alternatively, it can be seen as:
\[ v = \frac{\text{coefficient of } t}{\text{coefficient of } x} \]
Step 2: Detailed Explanation:
Given wave equation: \(y = 0.1 \sin(200t - 10x)\).
By comparing it with \(y = A \sin(\omega t - kx)\), we extract:
\(\omega = 200 \text{ rad/s}\)
\(k = 10 \text{ m}^{-1}\)
Substituting into the velocity formula:
\[ v = \frac{200}{10} = 20 \text{ m/s} \]
Since the signs of \(\omega t\) and \(kx\) are opposite, the wave is travelling in the positive \(x\)-direction.
Step 3: Final Answer:
The wave velocity is \(20 \text{ m/s}\).