Waves - Result Question 14

14. A transverse wave is represented by the equation $y=y_0 \sin \frac{2 \pi}{\lambda}(v t-x)$

For what value of $\lambda$ is the maximum particle velocity equal to two times the wave velocity?

[1998]

(a) $\lambda=2 \pi y_0$

(b) $\lambda=\frac{\pi y_0}{3}$

(c) $\lambda=\frac{\pi y_0}{2}$

(d) $\lambda=\pi y_0$

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Answer:

Correct Answer: 14. (d)

Solution:

  1. (d) $y=y_0 \sin \frac{2 \pi}{\lambda}(v t-x)$

Particle velocity,

$\frac{d y}{d t}=y_0 \times \frac{2 \pi}{\lambda} v \cos \frac{2 \pi}{\lambda}(v t-x)$

Maximum particle velocity $=y_0 \times \frac{2 \pi \nu}{\lambda}$

Wave velocity $=v \quad$ [given]

So, $y_0 \times \frac{2 \pi v}{\lambda}=2 v$

$\lambda=\pi \cdot y_0$