Standing Waves on a String
The notion of resonance can be extended to wave phenomena. Resonances in a wave medium (such as on a string or in the air, for sound) are standing waves; they are analogous to the resonant oscillation of a mass and spring. Unlike the mass and spring which has only a single resonant frequency, a stretched string has many frequencies that are resonant. These frequencies are called the harmonic series and are responsible for the generation of the pleasing tones from a piano, guitar, violin, or other stringed instruments. When we transfer energy to the strings of these instruments, they oscillate at the special frequencies determined by the harmonic series. When we watch the string move when it vibrates at one of the frequencies of the harmonic series, there is a standing wave pattern that is different for each frequency within the harmonic series.
Standing waves are intimately related to the other wave phenomena we have already discussed. We interpret standing waves as a superposition, or sum, of two traveling waves moving in opposite directions along the string. The traveling waves are reflected at the places where the string is firmly held. Since the string is held fixed at the end points, remember that positive wave pulses are reflected back as negative pulses.
The lowest standing wave frequency is called the fundamental or first harmonic. For this mode, all parts of the string vibrate together, up and down. Of course, the ends of the string are fixed in place and are not free to move. We call these positions nodes: a node is a point on the string that does not move. As we move along the string, the amplitude of oscillations at each position we look at changes, but the frequency of oscillation is the same. Near a node, the oscillation amplitude is very small.
In the middle of the string, the oscillation amplitude is largest; such a position is defined as an antinode. We assign a wavelength to the fundamental (and each higher harmonic discussed below) standing wave. At a fixed moment in time, all we observe is either a crest or a trough, but we never observe both at the same time for the lowest frequency standing wave. From this, we determine that half a standing wave length fits along the length of the string for the fundamental. Alternatively, we say that the wavelength of the fundamental is twice the length of the string, or
As we’ll discuss later, the oscillation frequencies of stretched strings effect the tone of the sounds we hear from instruments such as guitars, violins and cellos. Higher frequency oscillations result in higher-pitched tones; lower frequency oscillations produce lower-pitched tones. So how can we change the oscillation frequency of a stretched string? The above equation tells us. If we either increase the wave speed along the string or decrease the string length, we get higher frequency oscillations for the first (and higher) harmonic. Conversely, reducing the wave speed or increasing the string length lowers the oscillation frequency. How do we change the wave speed? Keep in mind, it is a property of the wave medium, so we have to do something to the string to alter the wave speed. From earlier discussions, we know that tightening the string increases the wave speed. We also know that more massive strings have smaller wave speeds.
As an example of how standing waves on a string lead to musical sounds, consider the first harmonic of a G string on a violin. The string is typically made of nylon, having a density of ~1200 kg/m3. The diameter of the G string is 4 mm. The string is held with a tension of 220 N. The frequency of the first harmonic of the G string is 196 Hz. What is the length of the string?
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. 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. 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. The vibration of the string is a superposition of standing waves, each with a different amplitude. The standing wave patterns of each mode are as we have shown above.