String instruments
Strings can be struck by a hammer (piano), plucked (guitar) or played by a bow (violin). The sound they generate are distinctly different, even though the basic idea for all three is that of a string with two fixed ends. This page should help us to explore the differences and understand the differences in sound.
The plucked String
The string receives its original energy in form of potential energy by being displaced from equilibrium and let go from rest.
We have shown that the envelope of the ensuing oscillation resembles the original shape of the displaced string, so every information about the wave is in the original shape. We can construct the envelope in terms of harmonic sine functions. Assume x to be the displacement of the string at any point along the string. Its motion can be described by
The coefficients An decay as 1/n with increasing frequency of the harmonics. Also, due to damping, the string will eventually have lost all its energy and return to the equilibrium position at rest. The sequence of sounds below builds the sound of a guitar string using these considerations:
440 Hertz, all harmonics in 1/n amplitude
440 Hertz, all harmonics in 1/n amplitude, every 5th removed
440 Hertz, all harmonics in 1/n amplitude, every 5th removed, damped
440 Hertz, all harmonics in 1/n amplitude, every 5th removed, damped and modulated
440 Hertz, all harmonics in 1/n amplitude, every 5th removed, frequency-dependent damping, modulated
The struck string
As can be found in a piano, some strings are played by being struck with a felt-tipped hammer, for example. Such strings receive their energy in form of an initial kinetic energy while at equilibrium position.
There is a physical difference in that the envelope of the wave on the string is now given as a distribution of initial velocity along the string.
The coefficients Bn are related to the coefficients An of the displacements as in
Therefore, the complex wave as the displacement of the string will have coefficients for its harmonic components, that decay faster than for the plucked string, namely as 1/n2. There also will be damping over time.
Using these pieces of information, the following sequence of sounds tries to piece it together to arrive at the sound by a piano string.
440 Hertz, all harmonics, 1/n2 amplitude
440 Hertz, all harmonics, 1/n2 amplitude, every 9th harmonic removed
440 Hertz, all harmonics, 1/n2 amplitude, every 9th harmonic removed, equally damped
The bowed string
The process of bowing produces a continuous sound, so the factor of damping will not play as much of a role in the formation of the sound as in the plucked and struck strings. When the bow is moving laterally across the string we observe basically a slip and slide process, in which the string gets picked up by the bow, displaced from equilibrium, then snaps and return a little beyond equilibrium and gets picked up again. As a result, the spectrum will follow a generalized saw tooth. The duty cycle is determined by the position of the bow along the string. Due to the interaction with the bow, the motion of the string does not resemble the motion of the plucked string, but more the one of the piano string with its 1/n2 dependency of the amplitudes An.
440 Hertz, all harmonics in 1/n amplitude
440 Hertz, all harmonics in 1/n amplitude, every 5th removed
440 Hertz, all harmonics in 1/n amplitude, every 5th removed, modulated
440
Hertz, all harmonics in 1/n2 amplitude, every 5th
removed, modulated