String Plucker

------ The String Plucker ------

(A uniform string, fixed at both ends, is plucked)
by M. Gallant 4/1996 5/2026

This animation shows the vibration of an ideal string with uniform tension and density, plucked at any position and oscillating with small amplitude and no energy loss.

Hold the mouse or touchpad down anywhere in the upper half of the frame below; move the mouse around with button down to set the initial "plucked position" and release the mouse to let the vibrating string evolve in time. Click in the field at any time to pause the animation. Hold the mouse down and drag to a new location to start a new animation. Toggle from Single to MultiPlot mode for cumulative images. Change the range of harmonic spatial/temporal frequencies in the display. Set the speed of the animation with the slider or text box. T is the fundamental period of oscillation and f₀ is the corresponding frequency of the displayed string animation. Watch and listen to the interesting visual and audio patterns as the plucked point is moved along the string.
(This modern JavaScript version, which runs on any modern browser supporting HTML5 Canvas and ES6 JavaScript, is based on the author's original 1996 Java Applet version, still available at The String Plucker ).

Speed: T = 1.00 s f0 = 1.00 Hz Q = 0.50

Harmonics:

Fundamental Audio Freq (Hz) fMax = 1960.00 Hz

Audio 196 Hz: Pure Tone, Center Pluck Q=0.5 , Pluck Q=0.25, Pluck Q=0.10
Audio frequency string vibrations can be studied using the audio synthesis and playback features using a selectable Fudamental Audio Frequency and the associated Harmonics Tour (showing each harmonic while playing the audio of that harmonic for 2 sec) and the AudioMix feature which, for a given plucked position, simulates the resultant composite sound vibration consisting of the harmonic components (see below for details).
This time animation is constructed from the exact Fourier series representation for an ideal (no damping) plucked string, vibrating in a plane, with uniform tension and an initial triangular profile as set by the user. The accuracy of this simulation is set by choosing the number of Fourier components ("harmonics") with the More and Less buttons. Pulling the string too close to one end will not faithfully reproduce the exact shape in time unless many components are used, requiring long computation times. The MultiPlot button allows plotting of multiple frames at fixed time intervals, representing the average string displacement noticeable to the eye, or in time-averaged photographs. The time evolution is constructed for one period T of the fundamental tone f₀ with a default value of T=1sec and a range of 0.1 to 10sec allowing for easy viewing of the vibration progression. Of course the actual value of T and f₀=1/T is determined by the physical parameters of the string. For example the parameters for a guitar unwound g-string that determine the vibration properties are:

String Length: 65 cm	String Linear Density: 1.6 gm/m
String Tension: 104 Newtons	Wave Velocity: 255 m/s
Fundamental Frequency: 196 Hz.	Fundamental Period: 5.1 ms

Since the frequency of each harmonic is an integral multiple of the fundamental frequency, the motion of the entire string, no matter where it is plucked, must repeat exactly after the fundamental period. The lower section of the panel contains a bar-graph profile of the Fourier amplitudes for the fundamental and higher harmonic components of the vibrating string. Values are normalized relative to the fundamental frequency. Unfilled bars correspond to negative amplitudes. The power in each harmonic is proportional to the square of the corresponding amplitude. Note that as the string is plucked closer to the ends, the high-frequency content increases, as expected. Note also that plucking the string at certain locations can result in suppression of a whole set of harmonics.

Analysis

An ideal string plucked with a small amplitude satisfies the classical one-dimensional wave equation, a linear second-order partial differential equation.

A plucked string of length L, displaced at position x_o by an amount y_o, can be expressed as a Fourier sine series:

$$ y(x,t) = \sum_{n=1}^{\infty} a_n\, \sin\!\left(\frac{n\pi x}{L}\right) \cos\!\left(2\pi n f_0 t\right) $$

Fourier Coefficients

$$ a_n = \frac{2\,y_0}{n^2 \pi^2\,Q(1 - Q)} \sin(n\pi Q), \qquad Q = \frac{x_0}{L} $$

Wave Speed, Frequency, and Period

$$ v = \sqrt{\frac{\text{Tension}}{\rho}}, \qquad f_0 = \frac{v}{2L}, \qquad T = \frac{1}{f_0} $$

Meaning of y_o

In the plucked string initial condition, the string is displaced into a triangular shape with its peak at the pluck point x_o. The parameter y_o is simply the maximum vertical displacement of the string at that peak. It sets the overall amplitude of the vibration but does not affect the relative strengths of the harmonics (the timbre). Changing y_o only scales the loudness uniformly.

Simplified Spectral Shape

Although the full Fourier coefficient contains several factors, only the terms that depend on \(n\) determine the timbre. Since y_o and Q(1-Q) do not depend on n, the spectral shape for a fixed pluck position Q is governed entirely by

$$ \text{shape}(n;Q) \propto \frac{\sin(n\pi Q)}{n^2}. $$

The factor sin(nπQ) encodes the pluck-position coloration (which harmonics are emphasized or suppressed), while the 1/n² dependence produces the characteristic triangular-displacement roll-off of higher harmonics. This compact expression captures the essential timbre of a plucked string at any fixed pluck point.

References

F.S. Crawford, Jr., Waves, 1968 McGraw-Hill, p. 48.
Horace Lamb, The Dynamical Theory of Sound, 1925 E. Arnold (1960 Dover), p. 72.
J.W.S. Rayleigh, The Theory of Sound, Vol. 1, 1894 Macmillan (1945 Dover), p. 184.