# Fourier Series Applet of 1D wave Motion, Plucked String.

Cuthbert Nyack
Fourier Series can be used to help find solutions to partial differential equations with boundary conditions. Here the 1D wave equation is considered. The PDE is shown below and assumed to apply to wave motion which can be anything from a string to electrons in a deep potential well. Here we assume a string of lenght L plucked at point pL
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This equation can be solved with boundary conditions. If the string is plucked at L/4 the BC are
y(x,0) = 4x/L for 0 £ x £ L/4 and y(x,0) = -4x/3L +4/3 for L/4 £ x £ L
y(0,t) = y(L,t) = 0 for t ³ 0.
With these conditions the solution can be obtained by separation of variables and is shown below.
The applet illustrates the vibration of the string when the string is plucked at 1 to 4 points(x1, x2, x3 and x4 with displacements a1, a2, a3, and a4).
Fn = 1 shows the string plucked at 1 point. Fn = 2 to 22 shows a special case of the initial displacement and resulting vibration of the string at different times.

Fn = 23 shows the string plucked at 2 points.
Fn = 24 to 45 shows a special case of the initial displacement and resulting vibration of the string.
Fn = 46 shows the string plucked at 3 points.
Fn = 47 to 67 and 68 to 88 shows special cases of the initial displacement and resulting vibration of the string.
Fn = 89 shows the string plucked at 4 points.
Fn = 90 to 100 show special cases of the initial displacement and resulting travelling waves on the string.
Fn = 101 to 119, 120 to 138, 139 to 157 shows special cases of the initial displacement and resulting vibration of the string.
Fn = 158 shows the string plucked at 1 point.
Fn = 159 to 170 show the vibration of the fundamental and first harmonic.
Fn = 171 shows the string plucked at 2 points.
Fn = 172 to 192 show the vibration of the fundamental and first harmonic.
Fn = 193 shows the string plucked at 3 points.
Fn = 172 to 201 show the vibration of the fundamental and first harmonic.
Fn = 202 shows the string plucked at 4 points.
Fn = 203 to 213 show the vibration of the fundamental and first harmonic.