Sound and Music - Mission SM6 Detailed Help

Diagram A shows the standing wave pattern created in a 95-cm long guitar string when it is vibrated at 240 Hz. Determine the vibrational frequency (in Hertz) that would be required of the same guitar string 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 if needed.
  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 guitar string is clamped at both of its ends. When the string is plucked, the ends are unable to vibrate. The ends then become nodes - points of no disturbance. Each consecutive node must be separated by an antinode. So for the lowest possible frequency (fundamental or first harmonic), there must be one antinode 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 nodes (on the ends) and one antinode, then the second harmonic has three nodes (two of which are on the ends of the string) and two antinodes. The third harmonic has four nodes and three antinodes. The fourth harmonic has five nodes and four antinodes. The fifth harmonic has ... - and so on.

The nth harmonic frequency (fn) of a set of natural frequencies is ntimes 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.