Question

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 statemen tis 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
• D. The frequency of the fundamental mode of vibrations is 137.5 Hz

Answer 1

Answer: (D)

Given speed of sound
$\upsilon=300 \,ms^{-1} $
And wave equation is
$y=y_{0} \sin \left(\frac{2\pi}{L}x\right).\sin \left(\frac{2\pi}{L}x+\frac{\pi}{4}\right)$
So, angular wave number,
$k=\frac{2\pi}{\lambda}=\frac{2\pi}{L}$
$\therefore \lambda=L=1.2\, m$
Frequency of fundamental vibration is
$v=\frac{\upsilon}{\lambda}=\frac{300}{1.2}=250\, Hz$