Question

A string of length $0.4 \,m$ and mass $10^{-2} kg$ is tightly clamped at its ends. The tension in the string is $1.6 \,N$. Identical wave pulses are produced at one end in equal intervals of time $\Delta t$. The minimum value of $\Delta t$, which allows constructive interference between successive pulses, is :

• A. 0.05 s
• B. 0.40 s
• C. 0.20 s
• D. 0.10 s

Answer 1

Answer: (D)

When a wave pulse is reflected from other end, it suffers a phase change of $\pi$ radians.
The speed of transverse wave in a flexible stretched string depends upon the tension $T$ in the string and the mass per unit length $(m)$ as
$v=\sqrt{\frac{T}{m}}=\sqrt{\frac{T}{m / l}}$
Putting the numerical values from the question,
$\therefore v =\sqrt{\frac{1.6}{10^{-2} / 0.4}} $
$=8 \,m / s$
We know that when pulse is reflected from the other end, it suffers a phase change of $180^{\circ}$. Again when the reflection takes place from first end, a phase change of $180^{\circ}$ in addition occurs. Therefore, after successive reflections same wave pulse is obtained as previous. If at this instant another same wave pulse is obtained, then these two wave pulses produce constructive interference.
Hence, for constructive interference between successive pulses
$\Delta t_{\min }=\frac{0.4+0.4}{8}=0.10 \,s$