Question

One end of a taut string of length $3m$ along the $x$-axis is fixed at $x=0$. The speed of the waves in the string is $100m/s$. The other end of the string is vibrating in the $y$-direction so that stationary waves are set up in the string. The possible waveform(s) of these stationary waves is (are)

• A. $y\left(t\right)=A\sin \left(\frac{2\pi x}{6}\right)\cos \left(\frac{50\pi t}{3}\right)$
• B. $y\left(t\right)=A\sin \left(\frac{\pi x}{6}\right)\cos \left(\frac{100\pi t}{3}\right)$
• C. $y\left(t\right)=A\sin \left(\frac{5\pi x}{6}\right)\cos \left(\frac{255\pi t}{3}\right)$
• D. $y\left(t\right)=A\sin \left(\frac{5\pi x}{6}\right)\cos \left(250\pi t\right)$

Answer 1

The correct option is D

$y\left(t\right)=A\sin \left(\frac{5\pi x}{6}\right)\cos \left(250\pi t\right)$

Step 1. Given data

1. The fixed end is a node while the free end is an antinode.
2. $x=0$ is a node
3. $x=3m$ is an antinode

Step 2. Formula used

1. Possible modes of vibration are $\mathrm{L}=\frac{\left(2\mathrm{n}+1\right)\mathrm{\lambda }}{4}$
2. Wave number $k=\frac{2\pi }{\lambda }$
3. Angular velocity $\omega =vk$
4. Equation of a waveform is in the form of $y\left(t\right)=A\sin \left(kx\right)\cos \left(\omega t\right)$

Where $\lambda$ is the wave length, $\omega$ angular velocity, $v$ wave speed, $t$ is the time and $k$ is the number of waves.

Step 3. Find the values of $k$ and $\omega$

$\mathrm{L}=\frac{\left(2\mathrm{n}+1\right)\mathrm{\lambda }}{4}$ where $n=0,1,2,3,...$

$$\begin{gathered} \Rightarrow \mathrm{\lambda }=\frac{4\mathrm{L}}{2\mathrm{n}+1} \\ =\frac{12}{2\mathrm{n}+1}\left[\because \mathrm{L}=3m\right] \end{gathered}$$

$k=\frac{2\pi }{\lambda }=\frac{\left(2n+1\right)\pi }{6}$

$\omega =vk=\frac{100\left(2n+1\right)\pi }{6}=\frac{50\left(2n+1\right)\pi }{3}$

For $$\begin{gathered} n=0,k=\frac{\pi }{6},\omega =\frac{50\pi }{3} \\ n=1,k=\frac{5\pi }{6},\omega =\frac{250\pi }{3} \\ ··· \\ ··· \\ n=7,k=\frac{5\pi }{2},\omega =250\pi \\ ··· \\ ··· \end{gathered}$$

and so on

Step 4: Find the possible waveform(s) of these stationary waves

For $n=0$

$y\left(t\right)=A\sin \left(\frac{\pi x}{6}\right)\cos \left(\frac{50\pi t}{3}\right)$

For $n=2$

$y\left(t\right)=A\sin \left(\frac{5\pi x}{6}\right)\cos \left(\frac{250\pi t}{3}\right)$

For $n=7$

$y\left(t\right)=A\sin \left(\frac{5\pi x}{2}\right)\cos \left(250\pi t\right)$

Hence, option D is correct.