Sound and Music - Mission SM8 Detailed Help

Diagram A shows the standing wave pattern created in a 95-cm long open-end air column when it is vibrated at 240 Hz. Determine the vibrational frequency (in Hertz) that would be required of the same air column to produce the standing wave pattern shown in Diagram B.

Natural Frequencies and Harmonics:
Every object has a natural frequency or a set of natural frequencies at which it tends to vibrate at. When struck, plucked, strummed or somehow disturbed, the object will vibrate at one of the natural frequencies in its set of natural frequencies. These individual frequency values are often referred to as the harmonic frequencies of the string or air column. The lowest harmonic frequency is referred to as the fundamental frequency. The other frequency values in the set of natural frequencies are whole number multiples of the fundamental frequency value.

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.

At the open ends of an air column, air is free to vibrate back and forth. Thus, the open ends are vibrational antinodes. When the air column is forced to resonate, the ends become antinodes. Each consecutive antinode must be separated by a node. So for the lowest possible frequency (fundamental or first harmonic), there must be one node between the two ends. 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 two antinodes (on the ends) and one node, then the second harmonic has three antinodes (two of which are at the ends of the air column) and two nodes. The third harmonic has four antinodes and three nodes. The fourth harmonic has five antinodes and four nodes. The fifth 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 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 second harmonic frequency (f2) can be determined by substituting 2 into the above equation in place of n.


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