Stationary Waves in a Closed Organ Pipe

Stationary Waves in a Closed Organ Pipe

Musical sounds can be produced by oscillating strings, air columns, membrane, steel bars and many other oscillating bodies. In most of the instruments, more than a single part take part in oscillation such as strings and body f violin vibrate producing musical sound.

Organ Pipe

A hollow wooden or metallic tube used to produce sound is called an organ pipe. It is wind instrument such as a flute, whistle, violin, clarinet etc. their air column in it set into vibrations by blowing air into it from one end. If both ends of the pipe are open, it is called an open organ pipe; flute is an example pipe.

Stationary Waves in a Closed Organ Pipe

Consider a closed organ pipe of length L as shown in the figure (a). A blast of air is blown into it at the open end and a wave thus travels through the pipe and is reflected at the next end. Due to a superposition of incident and reflected waves, stationary waves are produced. In the simplest mode of vibration, there is a displacement node, N at the closed end air is at rest there and a displacement antinode, An at the open end as the air can vibrate freely.

Fundamental Mode

In this mode of vibration, the pipe has one node at its closed end and one antinode at its open end. As observed in figure (a), the length, L of the pipe is equal to distance between a node and an antinode which is λ/4 where λ is the wavelength of the stationary wave. Then

$$\begin{align*} L &= \frac {\lambda }{4} \\ \text {or,} \: \lambda &= 4L \\ \text {If v is the velocity of sound and} \: f_1 \: \text {is the frequency of vibration,} \\ \text {we have} \\ v &= \lambda f_1 \\ \text {or,} \: f_1 &= \frac {v}{\lambda } = \frac {v}{4L} \\ \therefore f_1 &= \frac {v}{4L} \dots (i)\\ \end{align*}$$

It is the lowest frequency produced in the pipe. It is called fundamental frequency or first harmonic.

Overtones in Closed Pipe

If a stronger blast of air is blown into the pipe, the notes of higher frequencies are obtained. The two nodes of vibration in the same pipe as shown in the figure which is called the first overtone and second overtone.

Second Mode (first overtone)

In this mode of vibration, there are two antinodes and two nodes within the pipe. If λ is the wavelength of the wave, then we have

$$\begin{align*} L &= \frac {3\lambda }{4} \\ \text {or,} \: \lambda &= \frac {4L}{3} \\ \text {The frequency of vibration,} \: f_2 \: \text {is given by} \\ v &= \lambda f_2 \\ \text {or,} \: f_2 &= \frac {v}{\lambda } = \frac {v}{4L/3} = \frac {3v}{4L} = 3f_1 \\ \therefore f_2 &= 3f_1 \dots (ii) \\ \end{align*}$$

This is the frequency of first overtone or third harmonic. The frequency of the first over tone is three times the fundamental frequency.

Third Mode (Second Overtone)

In this node of vibration, there are three nodes and three antinodes within the pipe. If λ is wavelength of the wave, then

$$\begin{align*} L &= \frac {5\lambda }{4} \\ \text {or,} \: \lambda &= \frac {4L}{5} \\ \text {If} \: f_3 \: \text {is frequency of second overtone, then} \\ v &= \lambda f_3 \\ \text {or,} \: f_3 &= \frac {v}{\lambda } = \frac {v}{4L/5} = 5\frac {v}{4L} = 5f_1 \\ \therefore f_3 &= 5f_1 \dots (iii) \\ \end{align*}$$

This is the frequency of second overtone or fifth harmonic which is five times the fundamental frequency.

In this way, other higher modes of vibration can be obtained. From equations (i), (ii) and (iii), it is observed that the frequency of higher modes of vibration is odd integral multiples of fundamental frequency f₁. That is,

$$\begin{align*} \: f:f_1, \: 3f_1, \: 5f_1, \: 7f_1, \dots \dots \end{align*}$$

Conclusions

1. The frequencies of various harmonics are odd integral multiples of fundamental frequency.
2. The frequency of the first overtone is three times the fundamental frequency, frequency of the second overtone is five times the fundamental frequency. So the frequency of n^th overtone is (2n + 1) times the fundamental frequency.
3. The even harmonics are missing.
4. Since only odd harmonics are present, the sound quality is poor.