Question

The displacement of a progressive wave is given by \( y = 0.5 \sin(100t - 2x) \), where x and y are in meters and t is in seconds. The velocity of the wave is:

• A. \( 25\,\text{m s}^{-1} \)
• B. \( 50\,\text{m s}^{-1} \)
• C. \( 100\,\text{m s}^{-1} \)
• D. \( 200\,\text{m s}^{-1} \)

Hint:

For \( \sin(at - bx) \), wave speed is always: \[ v = \frac{a}{b} \]

Answer 1

The Correct Option is B

Step 1: Read the wave equation.
The wave is $y = 0.5\,\sin(100t - 2x)$. We match it to the standard form $y = A\,\sin(\omega t - kx)$ to read off its parts.

Step 2: Pick out the angular frequency.
The number in front of $t$ is the angular frequency, so $\omega = 100$ rad/s.

Step 3: Pick out the wave number.
The number in front of $x$ is the wave number, so $k = 2$ per metre.

Step 4: Recall the speed formula.
The speed of a travelling wave is the angular frequency divided by the wave number: $v = \dfrac{\omega}{k}$.

Step 5: Put the values in.
\[ v = \frac{100}{2}. \]

Step 6: State the answer.
Dividing gives the wave speed. \[ \boxed{50 \ \text{m s}^{-1}} \]