


2.4 Harmonics and pure tones
To understand why the graphs of pressure functions for the same notes might have different shapes, we will find it helpful to study the vibration of a single stretched string  for example a violin or guitar string. The pressure function associated with the vibration of a single string is determined by the motion of the string. If we stretch the string so that its profile resembles a sine graph and then release it, the resulting vibration will produce a pure tone.
The following applet shows the motion of the string when its initial profile resembles one arch of a sine function.
This particular vibration is called the fundamental or first harmonic. Another example of a vibration that will produce a pure tone occurs if the initial profile of the string contains two arches of the sine curve, i.e., one complete period of a sine graph. The next applet depicts this vibration.
This vibration is called the second harmonic or first overtone.
The next applet displays a number of possible pure tone vibrations of the single string.
You may have noticed that the more sinusoidal oscillations present in the initial profile of the string, the faster it vibrates. In fact, if the fundamental frequency of the string is w hertz, the frequency of the second harmonic will be 2w hertz, and, in general, the frequency of the nth harmonic will be nw. If you did not notice the frequency increasing for the higher harmonics, return to the last applet and pay careful attention to the period of the oscillations.
Now we turn to consideration of the pressure functions associated with pure tone vibrations of the string. Such a function will have the same frequency as that of the vibration of the string. If the fundamental vibration of the string has frequency w, then the associated pressure function p_{1}can be described by



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