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

(A uniform string, fixed at both ends, is plucked)
by M. Gallant 4/96

Hold the mouse down anywhere in the upper half of the framed applet field; 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 interrupt the plot and start a new "run". Watch the interesting distribution of harmonics in the lower bar graph as the plucked point is moved along the string.

**** Java aware browsers would animate the blue string below ****

This time animation is constructed from the exact Fourier series for an ideal plucked string, vibrating in a plane, with uniform tension and 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 of the fundamental tone. The parameters correspond to those for a guitar unwound g-string with the following values :

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

If the user selects a point outside the upper plot region, the initial point defaults to a central point. 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 applet contains a bar-graph profile of the Fourier amplitudes for the fundamental and higher harmonic components of the vibrating string. 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. For an excellent and concise account of sound including plucked, hammered and bowed strings, see :
Horace Lamb The Dynamical Theory of Sound 1925 E. Arnold (1960 Dover) p. 72.
see also :
J.W.S. Rayleigh The Theory of Sound V-1 1894 Macmillan (1945 Dover) p. 184.
F.S. Crawford, Jr. Waves 1968 McGraw Hill p. 48.

---------- The analysis ----------

where y is the vertical displacement of the string at x, x is the distance along the string from the left fixed end, L is the length of the string, T is the string tension, rho is the string linear density, xp is the plucked point, and t is the time.
The Java applet source code
(by M. Gallant 4/14/96)