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

 (A uniform string, fixed at both ends, is plucked)
by M. Gallant 4/1996    5/2026
Speed: T = 1.00 s f₀ = 1.00 Hz Q = 0.50
Harmonics:
fMax = 1960.00 Hz

Audio 196 Hz: Pure Tone, Center Pluck Q=0.5 , Pluck Q=0.25, Pluck Q=0.10

Analysis

A plucked string of length L, displaced at position xo by an amount yo, 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

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