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A standing wave pattern is established in a closed-end air column as it vibrates with the pattern shown below. The air column has a length of 80 cm. This pattern represents the ___ harmonic; its wavelength is ____ cm.
(Note: Your numbers may be different than those shown here.)
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Natural Frequencies and Standing Wave Patterns:
A standing wave pattern is characterized by the presence of nodes and antinodes that are always present at the same position along the medium. The fundamental frequency or first harmonic has the smallest possible number of nodes and antinodes. The standing wave patterns for the other harmonics - third, fifth, seventh, etc. - have an increasing number of nodes and antinodes.
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A closed-end air column consists of a column of air that is open to the surrounding environment at one end and closed off (capped, covered, etc.) at the opposite end. At the open end, air is free to vibrate back and forth. Thus, the open end is a vibrational antinode. At the closed end, air is not free to vibrate back and forth. The closed end is a vibrational node. When the air column is forced to resonate, a standing wave is produced with an antinode at one end and a node at the other end. So for the lowest possible frequency (fundamental or first harmonic), there must be a single node and antinode. The standing wave patterns for the other harmonics have additional nodes and antinodes in comparison to the first harmonic). So if the first harmonic has one antinode and one node, then the third harmonic has two antinodes and two nodes. The fifth harmonic has three antinodes and three nodes. The seventh harmonic has four antinodes and four nodes. The ninth harmonic has ... - and so on.
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The fundamental frequency is the lowest possible frequency produced by a air column. It is the first harmonic. It is generally expected that the next highest frequency would be the second harmonic. But don't be fooled! A closed-end air column does not have a second, fourth, sixth, or any even-numbered harmonic. Because the second highest frequency is three times the frequency of the fundamental, it is referred to as the third harmonic. Closed-end air columns will only have odd-numbered harmonics.
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Length-Wavelength Relationship:
A standing wave pattern shows a unique relationship between the wavelength of the waves that create the pattern and a length measured along the medium between two points on the pattern. Every nodal position on the pattern is separated from the next adjacent nodal position by one-half of a wavelength. Similarly, every antinodal position on the pattern is separated from the next adjacent antinodal position by one-half of a wavelength.
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