3.4 Resonance

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 $L$ is set vibrating in any one of the normal modes. After sometime the oscillation gradually dies out. The motion is damped by dissipation of energy through elastic support at the end and by resistance of air to the motion.

We can supply energy into the system by applying a driving force with the help of a tuning fork.

If the driving frequency is equal to any natural frequency of the string, the string will vibrate at that frequency with larger amplitude. This phenomenon is known as resonance.

String can vibrate in any one of the normal modes, therefore resonance can occur at many different frequencies.

Resonance will occur if the distance $L$ is an integral multiple of $\frac{\lambda }{2}$.

Mathematically, $$\begin{equation} \begin{aligned} L = n\frac{\lambda }{2} \\ \lambda = \frac{{2L}}{n} \\ \frac{v}{f} = \frac{{2L}}{n} \\ f = \frac{n}{{2L}}\sqrt {\frac{T}{\mu }} \\\end{aligned} \end{equation} $$