Like any object, an air column has a set of frequencies at which it naturally vibrates. When forced into resonate or vibrate, the air column will vibrate at one of these frequencies. These frequencies are referred to as harmonics. The frequencies are related to one another by whole number ratios. For instance, the frequency of the 5th harmonic is 5 times larger than the first harmonic. And the frequency of the nth harmonic is n times larger than the frequency of the first harmonic.
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Name That Harmonic: Closed-End Air Columns - help9
There are three similar versions of this question. Here is one of those versions:
Version 1:
The fundamental or first harmonic frequency of a closed-end air column is 60 Hz. The same air column is capable of vibrating with several other standing wave patterns. Identify the standing wave pattern for the harmonic that has a frequency of 420 Hz.
Identifying the Harmonic Number
Every air column has a set of frequencies with which it naturally vibrates. These frequencies are called harmonic frequencies and each is associated with a standing wave pattern. The lowest-pitched frequency in the set is called the first harmonic or fundamental frequency. All other frequencies are whole-number multiples of the fundamental frequency. The frequency values follow the equation fn = n•f1 where n is the harmonic number, f1 is the first harmonic (or fundamental) frequency, and fn is the frequency of the first harmonic. You can use the two given values for f1 and fn in this question to determine the harmonic number.
Selecting the Standing Wave Pattern
Standing wave patterns have nodes and antinodes. The nodes are locations where the air never vibrates. The antinodes are locations where the air vibrates wildly. The standing wave pattern for the first harmonic has the longest possible wavelength of all the harmonics. Given the rule that there are vibrational nodes at the closed end and antinodes at the open-end, the longest wavelength would have one-quarter of a wavelength. Each successive harmonic after the first harmonic would have an additional node and antinode. This is equivalent to adding another one-half wave to the pattern.
A Useful Table
The information above can be organized into a table as shown:
| Harmonic | # of Nodes | # of Antinodes | # of Waves in Pattern |
|---|---|---|---|
| 1 | 1 | 1 | 1/4 wave |
| 3 | 2 | 2 | 3/4 waves |
| 5 | 3 | 3 | 5/4 waves |
| 7 | 4 | 4 | 7/4 waves |
| 9 | 5 | 5 | 9/4 waves |
| 11 | 6 | 6 | 11/4 waves |
| n | (n+1)/2 | (n+1)/2 | n/4 waves |
Picking the Correct Pattern
You can use the table above to determine the characteristics of the standing wave that you are looking for. You can be certain that it begins with a node at the closed end and has an antinode at the open end. You will need to tap through the choices until you find a pattern that meets these criteria.
Try this link to The Physics Classroom Tutorial for more help with the concept of standing wave patterns in Closed-end air columns:
Closed-End Air Columns