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.
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.
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:
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Last updated: 11 Oct 1999