Question

A transverse wave is represented by the equation $y=y_0 \sin\frac{2\pi}{\lambda}(vt-x)$ For what value of $\lambda$, is the maximum particle velocity equal to two times the wave velocity?

• A. $\lambda=\frac{\pi y_0}{2}$
• B. $\lambda=\frac{\pi y_0}{3}$
• C. $\lambda=2\pi y_0$
• D. $\lambda=\pi y_0$

Answer 1

Answer: (D)

The given equation of wave is
$y=y_{0} \sin \frac{2 \pi}{\lambda}(v t-x)$
Particle velocity $=\frac{d y}{d t}=y_{0} \cos \frac{2 \pi}{\lambda}(v t-x) \cdot \frac{2 \pi v}{\lambda}$
$\left(\frac{d y}{d t}\right)_{\max }=y_{0} \cdot \frac{2 \pi}{\lambda} v$
$\therefore y_{0} \cdot \frac{2 \pi}{\lambda} v=2 v $
or, $ \lambda=\pi y_{0} $