Question

In the given wave equation $y=0.05 \sin \frac{2\pi}{\lambda}(x-200t)\text{ m}$, the velocity of the wave (in $\text{ms}^{-1}$) is

• A. $2\sqrt{200}$
• B. 400
• C. $200\sqrt{2}$
• D. $2\sqrt{300}$
• E. 200

Hint:

In any wave equation of the form $y = f(ax \pm bt)$, the wave speed is always the coefficient of time divided by the coefficient of position: $v = \frac{b}{a}$. Here, $v = \frac{(2\pi/\lambda) \times 200}{2\pi/\lambda} = 200$.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The standard equation of a traveling wave can be written as $y = A \sin(kx - \omega t)$ or $y = A \sin \frac{2\pi}{\lambda}(x - vt)$, where $v$ is the wave velocity.

Step 2: Detailed Explanation:
The given equation is:
\[ y = 0.05 \sin \left[ \frac{2\pi}{\lambda}(x - 200t) \right] \]
Comparing this with the standard form $y = A \sin \left[ \frac{2\pi}{\lambda}(x - vt) \right]$:
The term $(x - 200t)$ corresponds to $(x - vt)$.
Therefore, $v = 200\text{ ms}^{-1}$.

Step 3: Final Answer:
The velocity of the wave is 200 ms$^{-1}$.