Sound and Music - Mission SM10 Detailed Help

Diagram A shows the standing wave pattern created in a 95-cm long closed-end air column when it is vibrated at 270 Hz. Determine the vibrational frequency (in Hertz) which would be required of the same air column to produce the standing wave pattern shown in Diagram B.
(Note: Your numbers may be different than those shown here.)

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.

Often times the development of an effective strategy is the most difficult part of a physics question. The strategy below should prove useful.
  1. Two standing wave patterns are shown. The frequency of the pattern on the left is given. Identify the harmonic number for this pattern. See Think About It section.
  2. As discussed in the Know the Law section, the frequency of each harmonic is some multiple of the first harmonic's frequency. If not already known, determine the frequency of the first harmonic. Use the equation in the Formula Frenzy section.
  3. The goal is to determine the frequency associated with the pattern on the right. Identify the harmonic number for the pattern on the right. See Think About It section.
  4. Having found the harmonic number for the pattern on the right, you should be able to determine its frequency. Use the equation in the Formula Frenzy section.

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. An analysis of the diagrams should allow you to determine the harmonic number associated with each pattern. The equation in the Formula Frenzy section can then be used to determine the unknown frequency.

The lowest possible frequency produced by a air column 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 first harmonic, it is referred to as the third harmonic. Closed-end air columns will only have odd-numbered harmonics - first, third, fifth, seventh, etc.

The nth harmonic frequency (fn) of a set of natural frequencies is n times the frequency of the fundamental or first harmonic frequency (f1).
fn= n •f1

where n is a whole number. The third harmonic frequency (f3) can be determined by substituting 3 into the above equation in place of n.


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