Physics > Superposition of Waves > 4.0 Longitudinal stationary wave in an organ pipe
Superposition of Waves
1.0 Introduction
2.0 Interference of Waves
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
3.0 Standing or Stationary 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
4.0 Longitudinal stationary wave in an organ pipe
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
5.0 Beats
6.0 Questions
4.2 Closed organ pipe
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
3.2 Vibrations in a stretched string
3.3 Melde's Experient
3.4 Resonance
4.2 Closed organ pipe
4.3 End correction
4.4 Resonance tube
4.5 Energy in a stationary wave
- In a closed organ pipe one end is closed and another end is open
- The closed end is always a displacement node or pressure antinode
- The open end is always a displacement antinode or pressure node. [Diagram of open organ pipe]
| Normal modes of an organ pipe | Pressure Diagram | Displacement diagram | Explanation | Wavelength | Frequency |
First mode or first harmonic or fundamental tone |
|
| $$\frac{\lambda }{4} = L$$$$\lambda = 4L$$$$\frac{v}{f} = 4L$$$$f = \frac{v}{{4L}}$$ | $$\lambda = 4L$$ | $$f = \frac{v}{{4L}}$$ |
Second mode or second harmonic or first overtone |
|
| $$\frac{{3\lambda }}{4} = L$$$$\lambda = \frac{{4L}}{3}$$$$\frac{v}{f} = \frac{{4L}}{3}$$$$f = \frac{{3v}}{{4L}}$$ | $$\lambda = \frac{{4L}}{3}$$ | $$f = \frac{3v}{{4L}}$$ |
$n^{th}$ mode or $n^{th}$harmonic or $(n-1)^{th}$ overtone |
|
| $$\frac{{\left( {2n + 1} \right)\lambda }}{4} = L$$$$\lambda = \frac{{4L}}{{\left( {2n + 1} \right)}}$$$$\frac{v}{f} = \frac{{4L}}{{\left( {2n + 1} \right)}}$$$$f = \frac{{\left( {2n + 1} \right)v}}{{4L}}$$ | $$\lambda = \frac{{4L}}{{(2n + 1)}}$$ | $$f = \frac{{(2n + 1)v}}{{4L}}$$ |
Note: If an open organ pipe of length $L$ is half submerged in water. Then it will become closed organ pipe of length $\frac{L}{2}$.
For an organ pipe of length $L$,$$\begin{equation} \begin{aligned} \frac{\lambda }{2} = L \\ \lambda = 2L \\ f = \frac{v}{{2L}} \\\end{aligned} \end{equation} $$
For an organ pipe fill with water, $$\begin{equation} \begin{aligned} \frac{\lambda }{4} = \frac{L}{2} \\ \lambda = 2L \\ f = \frac{v}{{2L}} \\\end{aligned} \end{equation} $$
Frequency will remain unchanged because the fundamental frequency of an open organ pipe is twice that of a close organ pipe of the same length.
