Standing Waves in Open End Air Columns


Q) A pipe open at both ends resonates at a fundamental frequency $\ f_{open}.$ When one end is covered and the pipe is again made to resonate, the fundamental frequency is $\ f_{closed}.$ Which of the following expressions describes how these two resonant requencies compare?

A. $\ f_{closed}=f_{open}$       B. $\ f_{closed}=1/2f_{open}$
C. $\ f_{closed}=2f_{open}$      D. $\ f_{closed}=3/2f_{open}$

Answer) B.

Q2) Balboa Park in San Diego has an outdoor organ. When the air temperature increases, the fundamental frequency of one of the organ pipes 
A. stays the same,       B. goes down,
C. goes up, or            D. is impossible to determine.

Answer) C.
The sound emitted by the organ pipes increases in frequency as they warms up because the speed of sound increases in the increasingly warmer air inside them.

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Sound Waves
As sound waves travel through the air in the pipe, they are reflected at each end and travel back through the pipe. (The reflection occurs even if an end is open, but the reflection is not as complete as when the end is closed.) If the wavelength of the sound waves is suitably matched to the length of the pipe, the superposition of waves traveling in opposite directions through the pipe sets up a standing wave pattern. The wavelength required of the sound waves for such a match is one that corresponds to a resonant frequency of the pipe. The advantage of such a standing wave is that the air in the pipe oscillates with a large, sustained amplitude, emitting at any open end a sound wave that has the same frequency as the oscillations in the pipe. This emission of sound is of obvious importance to, say, an organist.

Standing Waves in Two Open Ends
The simplest standing wave pattern that can be set up in a pipe with two open ends is shown in Fig. 1a. There is an antinode across each open end, as required. 

Figure 1. (a) The simplest standing wave pattern of displacement for (longitudinal) sound waves in a pipe with both ends open has an antinode (A) across each end and a node (N) across the middle. (The longitudinal displacements represented by the double arrows are greatly exaggerated.) (b) The corresponding standing wave pattern for (transverse) string waves.

There is also a node across the middle of the pipe. An easier way of representing this standing longitudinal sound wave is shown in Fig. 1b—by drawing it as a standing transverse string wave. The standing wave pattern of Fig. 1a is called the fundamental mode or first harmonic. 

Figure 2. Motion of elements of air in standing longitudinal waves in a pipe, along with schematic representations of the waves. In the schematic representations, the structure at the left end has the purpose of exciting the air column into a normal mode. The hole in the upper edge of the column ensures that the left end acts as an open end. The graphs represent the displacement amplitudes, not the pressure amplitudes. In a pipe open at both ends, the harmonic series created consists of all integer multiples of the fundamental frequency: $\ f_{1}, 2f_{1}, 3f_{1} $, . . . .

For it to be set up, the sound waves in a pipe of length L must have a wavelength given by $\ L = \lambda /2$, so that $\lambda =  2L$. Several more standing sound wave patterns for a pipe with two open ends are shown in Fig. 2 using string wave representations.The second harmonic requires sound waves of wavelength $\lambda = L$, the third harmonic requires wavelength $\lambda = 2L/3$, and so on.
More generally, the resonant frequencies for a pipe of length $\ L$ with two open ends correspond to the wavelengths

$\lambda = \frac{2L}{n}, ~~~~~for~~ n=1,2,3, . . . $

where $\ n$ is called the harmonic number. Letting $\ v$ be the speed of sound, we write the resonant frequencies for a pipe with two open ends as 

$f=\frac{v}{\lambda }=\frac{nv}{2L}, ~~~~~for~~ n=1,2,3, . . . $

It is interesting to investigate what happens to the frequencies of instruments based on air columns and strings during a concert as the temperature rises. The sound emitted by a flute, for example, becomes sharp(increases in frequency) as the flute warms up because the speed of sound increases in the increasingly warmer air inside the flute. The sound produced by a violin becomes flat(decreases in frequency) as the strings thermally expand because the expansion causes their tension to decrease.

Musical instruments based on air columns are generally excited by resonance. The air column is presented with a sound wave that is rich in many frequencies. The air column then responds with a large-amplitude oscillation to the frequencies that match the quantized frequencies in its set of harmonics. In many woodwind instruments, the initial rich sound is provided by a vibrating reed. In brass instruments, this excitation is provided by the sound coming from the vibration of the player’s lips. In a flute, the initial excitation comes from blowing over an edge at the mouthpiece of the instrument in a manner similar to blowing across the opening of a bottle with a narrow neck. The sound of the air rushing across the edge has many frequencies, including one that sets the air cavity in the bottle into resonance.

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