3.2 Vibrations in a stretched string

1.1 Principle of superposition

2.1 Relation between phase difference $\left( \phi \right)$ and path difference $\left( {\Delta x} \right)$
2.2 Interference of waves from coherent sources
2.3 Interference of waves from incoherent sources
2.4 Reflection and transmission of a wave
2.5 Motion of wave during reflection
2.6 Expression for the reflection and transmission of wave

3.1 Transverse stationary wave on a stretched string
3.2 Vibrations in a stretched string
3.3 Melde's Experient
3.4 Resonance

4.1 Open organ pipe
4.2 Closed organ pipe
4.3 End correction
4.4 Resonance tube
4.5 Energy in a stationary wave

Consider a string of length $L$ fixed at both ends. When we set-up sinusoidal wave on such a string it gets reflected from fixed ends.

Due to the superposition of two identical waves travelling in opposite direction transverse standing waves are produced on the string.

Note: The fixed points should be nodes and has to be permanently at rest.

Speed of wave in a stretched string is given by, $$v = \sqrt {\frac{T}{\mu }} $$ where,

$T:$ Tension in the string

$\mu:$ Mass per unit length of the string

Normal modes of a string	Diagram	No. of loops	Explanation	Wavelength	Frequency
First harmonic or First mode or Fundamental tone		$n=1$	$$\begin{equation} \begin{aligned} \frac{\lambda }{2} = L \\ \lambda = 2L \\ \frac{v}{f} = 2L \\ f = \frac{v}{{2L}} = \frac{1}{{2L}}\sqrt {\frac{T}{\mu }} \\\end{aligned} \end{equation} $$	$$\lambda = 2L$$	$$\begin{equation} \begin{aligned} f = \frac{v}{{2L}} \\ f = \frac{1}{{2L}}\sqrt {\frac{T}{\mu }} \\\end{aligned} \end{equation} $$
Second harmonic or Second mode or First overtone		$n=2$	$$\begin{equation} \begin{aligned} \lambda = L \\ \frac{v}{f} = L \\ f = \frac{v}{L} = \frac{1}{L}\sqrt {\frac{T}{\mu }} \\\end{aligned} \end{equation} $$	$$\lambda = L$$	$$\begin{equation} \begin{aligned} f = \frac{v}{L} \\ f = \frac{1}{L}\sqrt {\frac{T}{\mu }} \\\end{aligned} \end{equation} $$
Third harmonic or Third mode or Second overtone		$n=3$	$$\begin{equation} \begin{aligned} \frac{{3\lambda }}{2} = L \\ \lambda = \frac{{2L}}{3} \\ \frac{v}{f} = \frac{{2L}}{3} \\ f = \frac{{3v}}{{2L}} = \frac{3}{{2L}}\sqrt {\frac{T}{\mu }} \\\end{aligned} \end{equation} $$	$$\lambda = \frac{{2L}}{3}$$	$$\begin{equation} \begin{aligned} f = \frac{{3v}}{{2L}} \\ f = \frac{3}{{2L}}\sqrt {\frac{T}{\mu }} \\\end{aligned} \end{equation} $$
$n^{th}$ harmonic or $n^{th}$ mode or $(n-1)^{th}$ overtone		$n=n$	$$\begin{equation} \begin{aligned} \frac{{n\lambda }}{2} = L \\ \lambda = \frac{{2L}}{n} \\ \frac{v}{f} = \frac{{2L}}{n} \\ f = \frac{{nv}}{{2L}} = \frac{n}{{2L}}\sqrt {\frac{T}{\mu }} \\\end{aligned} \end{equation} $$	$$\lambda = \frac{{2L}}{n}$$	$$\begin{equation} \begin{aligned} f = \frac{{nv}}{{2L}} \\ f = \frac{n}{{2L}}\sqrt {\frac{T}{\mu }} \\\end{aligned} \end{equation} $$