The
oscillation of a vibrating string can be described by the
sum(
∑) of a set of
sine and
cosine equations, a
Fourier Series. The specific
equation to describe the movement of a string is a
derivative, or rather a differential solution, of this general equation for any string is:
y(x,t) = ∑∞m=1sin(mπx/L){Cmsin(mωt) + Dm cos(mωt)}
where ω = π(c/L)
For a bowed string has a kink that moves up one side of the string, jumps over the axis and goes the other direction. When this kink passes the bow, the string rapidly moves in the opposite direction from the bow's direction where the kink passes again and the string then moves at the bow's velocity up to the point where the kink returns. The series to describe this follows:
y(x,t) = (8A/π²)∑∞m=1sin(mπx/L) sin{m(π/2-ωt)}
A plucked string begins in a position where part of the string is elevated by some height(assuming the string at rest to be a horizontal axis). When released the string moves toward rest, then becomes the reflection of the shape it took when it was plucked on the opposing side of the axis. This action repeats back and forth over the axis, its mathematical description is as follows:
y(x,t) = (2AL²/π²d(L-d))∑∞m=1(1/m²)sin(mπd/L)sin(mπx/L)cos(mωt)
where d is the distance between 0 and the point where the string is plucked on the axis L.
A hammered string is unique here in that a point on the string, d, is given a velocity. The result is a wave that displaces the string to a height, reflects at the ends of the string, displacing the string the same amount in the opposite direction. The result is a square wave the moves back and forth along the string. The equation for this is as follows:
y(x,t) = (A/ω)∑∞m=1(1/m)sin(mπd/L)sin(mπx/L)sin(mωt)