A standing wave in a pipe with a length L=1.2 ,m is described by … - Vidyadip

MCQ

A standing wave in a pipe with a length $L=1.2 \,m$ is described by $y(x, t)=y_0 \sin [(2 \pi / L) x] \sin [(2 \pi / L) x+\pi / 4]$ based on above information, which one of the following statement is incorrect? (Speed of sound in air is $300 \,ms ^{-1}$ )

A
The pipe is closed at both ends
B
The wavelength of the wave could be $1.2 ]\,m$
C
There could be a node at $x=0$ and antinode at $x=L / 2$
✓
The frequency of the fundamental mode of vibrations is $137.5 \,Hz$

✓

Answer

Correct option: D.

The frequency of the fundamental mode of vibrations is $137.5 \,Hz$

d
(d)

Given speed of sound,

$v=300 \,ms ^{-1}$

And wave equation is

$y=y_0 \sin \left(\frac{2 \pi}{L} x\right) \cdot \sin \left(\frac{2 \pi}{L} x+\frac{\pi}{4}\right)$

So, angular wave number,

$k=\frac{2 \pi}{\lambda}=\frac{2 \pi}{L}$

$\therefore \quad \lambda=L=1.2 \,m$

Frequency of fundamental vibration is

$v=\frac{v}{\lambda}=\frac{300}{12}=250 \,Hz$

So, option $(d)$ is incorrect.

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