Standing Waves in Closed End Air Columns Simulation

Quiz

Q1) For a closed pipe, the second overtone is the

A. second harmonic
B. third harmonic
C. fourth harmonic
D. fifth harmonic

Answer) D.

Q2) The physical property that is most responsible for resonance is

A. waveform
B. frequency
C. quality
D. intensity

Answer) B.

Standing Waves in Closed End Air Columns Simulation

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Standing Waves In Air Columns

The waves under boundary conditions model can also be applied to sound waves in a column of air such as that inside an organ pipe. Standing waves are the result of interference between longitudinal sound waves traveling in opposite directions.

In a pipe closed at one end, the closed end is a displacement node because the rigid barrier at this end does not allow longitudinal motion of the air. Because the pressure wave is 90° out of phase with the displacement wave (see Section 17.2), the closed end of an air column corresponds to a pressure antinode (that is, a point of maximum pressure variation).

The open end of an air column is approximately a displacement antinode and a pressure node. We can understand why no pressure variation occurs at an open end by noting that the end of the air column is open to the atmosphere; therefore, the pressure at this end must remain constant at atmospheric pressure.

Figure 1. 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. (a) 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} $, . . . . (b) In a pipe closed at one end and open at the other, the harmonic series created consists of only odd-integer multiples of the fundamental frequency: $\ f_{1}, 3f_{1}, 5f_{1} $, . . . .

With the boundary conditions of nodes or antinodes at the ends of the air column, we have a set of normal modes of oscillation as is the case for the string fixed at both ends. Therefore, the air column has quantized frequencies.

The first three normal modes of oscillation of a pipe open at both ends are shown in Figure 1a. Notice that both ends are displacement antinodes (approximately). In the first normal mode, the standing wave extends between two adjacent antinodes, which is a distance of half a wavelength. Therefore, the wavelength is twice the length of the pipe, and the fundamental frequency is $\ f_{1} = v/2L $. As Figure 1a shows, the frequencies of the higher harmonics are $\ 2f_{1}, 3f_{1} $, . . . .

In a pipe open at both ends, the natural frequencies of oscillation form a harmonic series that includes all integral multiples of the fundamental frequency.

Because all harmonics are present and because the fundamental frequency is given by the same expression as that for a string, we can express the natural frequencies of oscillation as

$\ f_{n} = n \frac{v}{2L} ~~~~~~~n=1,2,3,... ~~~~~(1)$

Despite the similarity between Equations 18.5 and 18.8, you must remember that $\ v $ in Equation 18.5 is the speed of waves on the string, whereas $\ v $ in Equation 1 is the speed of sound in air.

If a pipe is closed at one end and open at the other, the closed end is a displacement node (see Fig. 1b). In this case, the standing wave for the fundamental mode extends from an antinode to the adjacent node, which is one-fourth of a wavelength.

Hence, the wavelength for the first normal mode is $\ 4L $, and the fundamental frequency is $\ f_{1} = v/4L $. As Figure 1b shows, the higher-frequency waves that satisfy our conditions are those that have a node at the closed end and an antinode at the open end; hence, the higher harmonics have frequencies $\ 3f_{1}, 5f_{1} $, . . . .

In a pipe closed at one end, the natural frequencies of oscillation form a harmonic series that includes only odd integral multiples of the fundamental frequency.

We express this result mathematically as

$\ f_{n} = n \frac{v}{4L} ~~~~~~~n=1,3,5,... ~~~~~(2)$

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 (consider Eq. 2).

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.