Open Organ Pipe and End Correction in Pipes

Stationary Waves in an Open Organ Pipe

An open pipe is one which is opened at both ends. When air was blown into the pipe through one end, a wave travels through the tube to the next end from where it is reflected. Due to a superposition of the incident and reflected waves, a stationary wave is set up in the air in the pipe.

Fundamental Mode (First Harmonic)

Let length of pipe be L and the velocity of sound in air is v. in this mode of vibration, there are antinodes at open end and one node at the middle of the pipe as shown in the figure. Let λ be the wavelength of the waves. Then,

$$\begin{align*} L &= \frac {\lambda }{2} \\ \text {or,} \: \lambda &= 2L \\ \text {Thus the frequency of fundamental mode or first harmonic is} \\ \text {or,} \: f_1 &= \frac {v}{\lambda } = \frac {v}{2L} \dots (i) \\ \end{align*}$$

Overtones in Open Organ Pipe

If a stronger blast of air is blown into the pipe, notes of higher frequencies are obtained which are called overtones. Two overtones are discussed below.

Second Mode (First Overtone)

In this mode of vibration, two antinodes are at the open ends but inside the pipe, there are two nodes and one antinode as shown in figure. If λ be the wave length of the wave, then

$$\begin{align*} L &= \lambda \\ \text {Thus, the frequency of first over tone or second harmonic is} \\ f &= \frac {v}{\lambda } = \frac {v}{L} = 2\frac {v}{2L} = 2f_1 \\ \therefore f_2 &= f_1 \dots (ii) \\ \end{align*}$$

Third Mode {Second Overtone)

In this mode of vibration, two antinodes are produced at both open ends and inside the pipe there are three nodes and two antinodes. If λ be the wave length of the wave then,

$$\begin{align*} L &= \frac {3\lambda }{2} \\ \text {or,} \: \lambda &= \frac {2L}{3} \\ \text {Thus, the frequency of second overtone or third harmonic is } \\ f_3 &= \frac {v}{\lambda } = \frac {3v}{2L} = 3 \frac {v}{2L} \\ \therefore f_3 &= 3f_1 \dots (iii) \\ \end{align*}$$

In this way, other higher modes of vibration can be obtained. From equation (i), (ii) and (iii), it is found that frequencies of higher modes of vibration are integral multiple of fundamental frequency f₁. So, all harmonics are possible in an open pipe.

Conclusions

1. The frequencies of various modes of vibration are an integral multiple of the fundamental frequency.
2. The frequency of the first overtone is two times the fundamental frequency; the frequency of the second overtone is three times the fundamental frequency. So the frequency of n^th overtone is (n + 1) times the fundamental frequency.
3. All harmonics are present.
4. Since sound produced by open end organ pipe contains all harmonics, so it is richer in quality than produced by closed end organ pipe.
5. The fundamental frequency of an open pipe is twice that of a closed of the same length.
6. The note from an open pipe is richer than that of a closed pipe owing to the presence of extra overtone.

End Correction in Pipes

In organ pipes, we take the antinode exactly at the open end and calculation of L is made accordingly. However, the antinode actually lies a little outside the open end because the air just outside the open end is set into vibration. So the displacement antinode of a stationary wave occurs a little distance beyond the end. The distance between the antinode and the open end of the pipe is called the end correction. The end correction, denoted by ‘e’, in a closed pipe as in figure is given by

$$\begin{align*} L + e &= \frac {\lambda }{4} \\ \text {or,} \: \lambda &= 4(L + e) \\ \text {where L is length of the pipe.} \\ \text {In case of an open organ pipe sounding in fundamental mode as in figure,} \\ \text {The end correction is given by} \\ L + e + e &= \frac {\lambda }{2} \\ \text {or,} \: L + 2e = \frac {\lambda }{2} \\ \text {or,} \: \lambda &= 2(L + 2e) \\ \end{align*}$$

The end correction is needed at two ends in an open pipe. The end correction determined mathematically is e = 0.58 r or 0.6 r ; where r is radius. The temperature and end correction affect on the frequency of sound in a pipe. For example, take a closed pipe sounding in a fundamental mode. The frequency of the mode is

$$\begin{align*} f &= \frac {v}{\lambda } = \frac {v}{4(L + e)} \\ \text {The velocity of sound v, at temperature} \: \theta \: \text {in terms of velocity} \: v_o \: \text {at} \: 0^o\: C\: \\\text {given by} \\ \frac {v}{v_o} &= \sqrt {\frac {T_o}{T}} \\ &= \sqrt {\frac {273 + \theta }{273}} \\ &= \sqrt {1 + \frac {\theta }{273}} \\ \text {Then,} f &= \frac {4}{4(L + e)} \\ &= \frac {v_o}{4(L + e)} \sqrt {1 + \frac {\theta }{273}} \\ \end{align*}$$

So, the frequency of fundamental mode increases as the temperature rises. It follows that for a given temperature and the length of pipe, the frequency decreases as e increases.