A 32 cm long string produces sound of frequency 400 Hz. Find the frequency of
sound if the length of the string is 24 cm.
Answers
Answer:
Recognizing the Length-Wavelength Relationship
First, consider a guitar string vibrating at its natural frequency or harmonic frequency. Because the ends of the string are attached and fixed in place to the guitar's structure (the bridge at one end and the frets at the other), the ends of the string are unable to move. Subsequently, these ends become nodes - points of no displacement. In between these two nodes at the end of the string, there must be at least one antinode. The most fundamental harmonic for a guitar string is the harmonic associated with a standing wave having only one antinode positioned between the two nodes on the end of the string. This would be the harmonic with the longest wavelength and the lowest frequency. The lowest frequency produced by any particular instrument is known as the fundamental frequency. The fundamental frequency is also called the first harmonic of the instrument. The diagram at the right shows the first harmonic of a guitar string. If you analyze the wave pattern in the guitar string for this harmonic, you will notice that there is not quite one complete wave within the pattern. A complete wave starts at the rest position, rises to a crest, returns to rest, drops to a trough, and finally returns to the rest position before starting its next cycle. (Caution: the use of the words crest and trough to describe the pattern are only used to help identify the length of a repeating wave cycle. A standing wave pattern is not actually a wave, but rather a pattern of a wave. Thus, it does not consist of crests and troughs, but rather nodes and antinodes. The pattern is the result of the interference of two waves to produce these nodes and antinodes.) In this pattern, there is only one-half of a wave within the length of the string. This is the case for the first harmonic or fundamental frequency of a guitar string. The diagram below depicts this length-wavelength relationship for the fundamental frequency of a guitar string.
The second harmonic of a guitar string is produced by adding one more node between the ends of the guitar string. And of course, if a node is added to the pattern, then an antinode must be added as well in order to maintain an alternating pattern of nodes and antinodes. In order to create a regular and repeating pattern, that node must be located midway between the ends of the guitar string. This additional node gives the second harmonic a total of three nodes and two antinodes. The standing wave pattern for the second harmonic is shown at the right. A careful investigation of the pattern reveals that there is exactly one full wave within the length of the guitar string. For this reason, the length of the string is equal to the length of the wave.
The third harmonic of a guitar string is produced by adding two nodes between the ends of the guitar string. And of course, if two nodes are added to the pattern, then two antinodes must be added as well in order to maintain an alternating pattern of nodes and antinodes. In order to create a regular and repeating pattern for this harmonic, the two additional nodes must be evenly spaced between the ends of the guitar string. This places them at the one-third mark and the two-thirds mark along the string. These additional nodes give the third harmonic a total of four nodes and three antinodes. The standing wave pattern for the third harmonic is shown at the right. A careful investigation of the pattern reveals that there is more than one full wave within the length of the guitar string. In fact, there are three-halves of a wave within the length of the guitar string. For this reason, the length of the string is equal to three-halves the length of the wave. The diagram below depicts this length-wavelength relationship for the fundamental frequency of a guitar string.