Digital Sound & Music: Concepts, Applications, & Science, Chapter 2, last updated 6/25/2013
is called a standing wave. In order to get the rope into this state, you have to shake the rope at a
resonant frequency. A rope can vibrate at more than one resonant frequency, each one giving
rise to a specific mode – i.e., a pattern or shape of vibration. At its fundamental frequency, the
whole rope is vibrating up and down (mode 1). Shaking at twice that rate excites the next
resonant frequency of the rope, where one half of the rope is vibrating up while the other is
vibrating down (mode 2). This is the second harmonic (first overtone) of the vibrating rope. In
the third harmonic, the “up and down” vibrating areas constitute one third of the rope‟s length
Figure 2.10 Vibrating a rope at a resonant frequency
This phenomenon of a standing wave and resonant frequencies also manifests itself in a
musical instrument. Suppose that instead of a rope, we have a guitar string fixed at both ends.
Unlike the rope that is shaken at different rates of speed, guitar strings are plucked. This pluck,
like an impulse, excites multiple resonant frequencies of the string at the same time, including
the fundamental and any harmonics. The fundamental frequency of the guitar string results from
the length of the string, the tension with which it is held between two fixed points, and the
physical material of the string.
The harmonic modes of a string are depicted in Figure 2.11. The top picture in the figure
illustrates the string vibrating according to its fundamental frequency. The wavelength of the
fundamental frequency is two times the length of the string L.
The second picture from the top in Figure 2.11 shows the second harmonic frequency of
the string. Here, the wavelength is equal to the length of the string, and is twice the frequency of
the fundamental. In the third harmonic frequency, the wavelength is 2/3 times the length of the
string, and three times the frequency of the fundamental. In the fourth harmonic frequency, the
wavelength is 1/2 times the length of the string, and four times the frequency of the fundamental.
More harmonic frequencies could exist beyond this depending on the type of string.