Question

A wave disturbance in a medium is described by $y(x,t)=0.02$ $\cos (50\pi t+\frac{\pi }{2})\cos \left(10\pi x\right)$, where $x$ and $y$ are in metre and $t$ in second

	Column A		Column B
1.	A displacement node occurs at $x=0.15m$	A.	$\left(i\right),\left(ii\right),\left(iii\right)$
2.	An antinode occurs at $x=0.3m$	B.	$\left(i\right),\left(ii\right),\left(iv\right)$
3.	The wavelength of the wave is $0.2m$	C.	$\left(ii\right),\left(iii\right),\left(iv\right)$
4.	The speed of the wave is $5.0m/s$	D.	all

• A. $a$
• B. $c$
• C. $d$
• D. $b$

Answer 1

The correct option is C $d$

Find speed of the wave.
Formula used: $v=\frac{\omega }{k}$
$y=0.02$ $\cos (50\pi t+\frac{\pi }{2})\cos \left(10\pi x\right)$
$y=-0.02\sin \left(50\pi t\right)\cos \left(10\pi x\right)\dots \left(i\right)$
As we know,
the standard equation of the wave is,
$y=-2A\sin \left(\omega t\right)\cos \left(kx\right)\dots \left(ii\right)$
From $\left(i\right)$ and $\left(ii\right)$,
$\omega =50\pi ,k=10\pi$
Wave speed $v=\frac{\omega }{k}=\frac{50\pi }{10\pi }=5m/s$, $\left(iv\right)$ is correct.

Find wavelength of the wave.
Formula used:
$k=\frac{2\pi }{\lambda }$
$k=\frac{2\pi }{\lambda }=10\pi$
$\Rightarrow \lambda =0.2m,\left(iii\right)$ is correct.

Find position of the nodes.
Formula used: $\cos \left(kx\right)=0$
For nodes,
$\cos \left(kx\right)=0=\cos \frac{\pi }{2},\cos \frac{3\pi }{2},\dots$
$x=\frac{\pi }{2k},\frac{3\pi }{2k},\dots =\frac{1}{20},\frac{3}{20},\dots$
$=0.05m,0.15m,\dots ,\left(i\right)$ is correct.

Find position of the antinodes.
Formula used: $\cos \left(kx\right)=\pm 1$
For antinodes, $\cos \left(kx\right)=\pm 1=\cos 0,\cos \pi ,\cos 2\pi$
$x=0,\frac{\pi }{k},\frac{2\pi }{k},\frac{3\pi }{k},\dots$
$=0,0.1m,0.2m,0.3m,\dots ,\left(ii\right)$ is correct.
Final answer: (𝑑)