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A string of mass per unit length $\mu$ is under tension $T$. The speed of a wave travelling on this string is given by \begin{align} v=\sqrt{\frac{T}{\mu}}. \end{align}

Two similar waves travelling in the opposite direction produces standing waves. The displacement of superposed wave is zero at the nodes and it is the maximum at the antinodes.

Consider a string of length $L$ fixed at the both ends. The wave displacement at the both ends is zero. This is given by boundary conditions: $y=0 \text{ at } x=0 \text{ and at } x=L$.

The frequencies allowed on this string are given by conditions \begin{align} L&=n\frac{\lambda}{2},\\ \nu&=\frac{n}{2L}\sqrt{\frac{T}{\mu}},\\ n&=1,2,3,\ldots. \end{align}

The fundamental or 1st harmonics is given by \begin{align} \nu_0=\frac{1}{2L}\sqrt{\frac{T}{\mu}} \end{align}

1st overtone of 2nd harmonics is given by \begin{align} \nu_1=\frac{2}{2L}\sqrt{\frac{T}{\mu}} \end{align}

2nd overtone of 3rd harmonics is given by \begin{align} \nu_2=\frac{3}{2L}\sqrt{\frac{T}{\mu}} \end{align}

All harmonics are present in a string fixed at the both ends.

Sonometer is a string fixed at the both ends. Following are useful relations for a sonometer \begin{align} \nu & \propto \frac{1}{L} \\ \nu & \propto \sqrt{T} \\ \nu &\propto \frac{1}{\sqrt{\mu}} \\ \nu & =\frac{n}{2L}\sqrt{\frac{T}{\mu}} \end{align}

Consider a string of length $L$ fixed at one end and free at the other end. The wave displacement at the fixed end is zero. This is given by boundary conditions: $y=0 \text{ at } x=0$.

The frequencies allowed on this string are given by conditions \begin{align} L & =(2n+1)\frac{\lambda}{4}\\ \nu &=\frac{2n+1}{4L}\sqrt{\frac{T}{\mu}} \\ n&=0,1,2,\ldots \end{align}

The fundamental or 1st harmonics is given by \begin{align} \nu_0=\frac{1}{4L}\sqrt{\frac{T}{\mu}} \end{align}

1st overtone of 3rd harmonics is given by \begin{align} \nu_1=\frac{3}{4L}\sqrt{\frac{T}{\mu}} \end{align}

2nd overtone of 5th harmonics is given by \begin{align} \nu_2=\frac{5}{4L}\sqrt{\frac{T}{\mu}} \end{align}

Only odd harmonics are present in a string fixed at one end.

**Question 1:**
A string is fixed at the two ends. The fundamental frequency is $\nu_0$. The string is plucked at the middle and then released. Mark the correct statement.

- The string will vibrate in fundamental mode with a unique frequency $\nu_0$.
- The vibration will be a superposition of modes with frequencies mainly $\nu_0, 2\nu_0, 3\nu_0$.
- The vibration will be a superposition of modes with frequencies mainly $\nu_0, 3\nu_0, 5\nu_0$.

**Question 2**
The figure shown is an actual photograph of a string vibrating in its fundamental mode. Explain why the edges of the figure are sharper then the middle portion.

**Problem (IIT JEE 2000): **
Two vibrating strings of the same material but of lengths $L$ and $2L$ have radii $2r$ and $r$ respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one of length $L$ with frequency $\nu_1$ and the other with frequency $\nu_2$. The ratio $\nu_1/\nu_2$ is given by,

- $2$
- $4$
- $8$
- $1$

**Solution: **
Let the lengths and radii of the two strings be $l_1=L$, $r_1=2r$, $l_2=2L$, and $r_2=r$, respectively. Let $\rho$ be the density of the material. The mass and mass per unit length of the first string are,
\begin{align}
m_1 &=\rho (\pi r_1^2 l_1) \\
&=4\pi \rho r^2 L,\\
\mu_1&=\frac{m_1}{l_1}=4\pi \rho r^2,
\end{align}
and that of the second string are,
\begin{align}
m_2&=\rho (\pi r_2^2 l_2) \\
&=2\pi \rho r^2 L, \\
\mu_2 & =\frac{m_2}{l_2}=\pi \rho r^2.
\end{align}
In fundamental mode, $l=\lambda/2=v/(2\nu)$ which gives,
\begin{align}
\nu=\frac{v}{2l}=\frac{1}{2l}\sqrt{\frac{T}{\mu}}.
\end{align}
Substitute $\mu$ from first and second equation to get,
\begin{align}
\nu_1&=\frac{1}{2l_1}\sqrt{\frac{T_1}{\mu_1}} \\
&=\frac{1}{2L}\sqrt{\frac{T}{4\pi\rho r^2}} \\
&=\frac{1}{4L}\sqrt{\frac{T}{\pi\rho r^2}},\nonumber\\
\nu_2 & =\frac{1}{2l_2}\sqrt{\frac{T_2}{\mu_2}} \\
&=\frac{1}{2(2L)}\sqrt{\frac{T}{\pi\rho r^2}} \\
&=\frac{1}{4L}\sqrt{\frac{T}{\pi\rho r^2}}.\nonumber
\end{align}