Mathematics of Standing Waves
of a stretched string is such that the wavelength is twice the length of the string. basic wave relationship gives an expression for the fundamental frequency. The relation between pulse speed, tension and linear density is given by the The wavelengths of the standing waves are fixed by the length of the string. The wavelength of the fundamental standing wave on a cello string depends on which of these quantities: length of the string, mass per unit length of the string.
These frequencies and their associated wave patterns are referred to as harmonics. A careful study of the standing wave patterns reveal a clear mathematical relationship between the wavelength of the wave that produces the pattern and the length of the medium in which the pattern is displayed.
Furthermore, there is a predictability about this mathematical relationship that allows one to generalize and deduce a statement concerning this relationship. To illustrate, consider the first harmonic standing wave pattern for a vibrating rope as shown below. Analyzing the First Harmonic Pattern The pattern for the first harmonic reveals a single antinode in the middle of the rope.
This antinode position along the rope vibrates up and down from a maximum upward displacement from rest to a maximum downward displacement as shown.
The vibration of the rope in this manner creates the appearance of a loop within the string. A complete wave in a pattern could be described as starting at the rest position, rising upward to a peak displacement, returning back down to a rest position, then descending to a peak downward displacement and finally returning back to the rest position. The animation below depicts this familiar pattern.
How do you calculate the wavelength of a standing wave?
As shown in the animation, one complete wave in a standing wave pattern consists of two loops. Thus, one loop is equivalent to one-half of a wavelength.
The string is held with a tension of N. The frequency of the first harmonic of the G string is Hz. What is the length of the string?
Higher harmonics Higher harmonics within the harmonic series come from successively adding nodes fixed points, where the string doesn't move to the standing wave pattern.
Every time there is an additional node, the frequency gets higher. The next frequency after the fundamental is known as the second harmonic. Instead of just the two nodes at the places where the string is held, we have added a third node, right in the middle of the string.
The standing wave pattern at an instant in time now has half the string moving downward while the other half moves upward. At later times, this pattern reverses. There is both a crest and a trough at any instant in time. This means that the wavelength of the second harmonic equals the length of the string.
We can again find the frequency of the second harmonic, by using the relationship between wave speed, wavelength and frequency.
Standing Waves on a String
As with all wave phenomena, the wave speed does not change with the frequency. It depends on the properties of the medium, alone. For the second harmonic In words, the second harmonic has twice the frequency of the fundamental.
Since the wave speed is the same for both standing waves, it also follows that the second harmonic has half the wave length as the fundamental. The higher harmonic standing waves are called overtones.
The second harmonic overtone can be easily heard on a guitar by laying your finger lightly on the string at the midpoint between the two frets after the string has been plucked. If you do this, you hear a faint, higher frequency tone. Successively higher harmonics are formed by adding successively more nodes. The third harmonic has two more nodes than the fundamental, the nodes are arranged symmetrically along the length of the string.
Mathematics of Standing Waves
One third the length of the string is between each node. The standing wave pattern is shown below. From looking at the picture, you should be able to see that the wavelength of the second harmonic is two-thirds the length of the string. We find the wavelength of the third harmonic from the standing wave pattern shown above; it is two-thirds of the length of the string.
We find the frequency of this mode: From the three detailed examples we did, I hope you can see a pattern in the standing waves on a string for the higher harmonics. The rules for the pattern are For each higher harmonic, we add a node to the standing wave pattern.
Between any two adjacent nodes, there is an antinode, where the oscillation amplitude is largest. All of the nodes are symmetrically placed along the length of the string. The frequency of the nth harmonic is the integer n times the fundamental frequency. This means that the fourth harmonic is four times higher in frequency than the fundamental, and so on.
When we somehow transfer energy to a string through a vibration, the harmonic series serves to filter out those vibrations occuring at the special frequencies, n f1.