Vibrating String Demo Chuck Gartland Department of Mathematics and Computer Science Kent State University April, 1997 This Maple worksheet produces animations of a vibrating string (with fixed end points), started from rest, with a prescribed initial displacement. It contrasts the truncated Fourier series solution (using a prescibed number of modes) with the exact solution, represented as a superposition of left/right traveling waves. Re-initialize workspace. restart; Include special plotting functions. with( plots ): Define an initial displacement function on (0,1). Uncomment one of the functions below, or create one of your own. Use the Heaviside function for piecewise-defined functions. hat function f := x -> x*Heaviside(x) + (1-2*x)*Heaviside(x-1/2);

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smaller hat function # f := x -> (x-1/3)*Heaviside(x-1/3) + (1-2*x)*Heaviside(x-1/2) + (x-2/3)*Heaviside(x-2/3); square pulse # f := x -> Heaviside(x-1/3) - Heaviside(x-2/3); constant function # f := x -> 1; parabolic function # f := x -> x * ( 1 - x ); semi-circle function # f := x -> sqrt( x * (1-x) ); fundamental mode # f := x -> sin( Pi*x ); your favorite function # f := x -> ??? Preview the initial displacement function. plot( f, 0..1, discont=true, color=red );

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Odd periodic extension of initial displacement function. F := proc(x) local s; s := x - 2 * floor( (x+1) / 2 ); signum( 0, s, 0 ) * f( abs(s) ) end: True solution. True := (x,t) -> ( F(x+t) + F(x-t) ) / 2;

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Generate (and save) animation plotting data for true solution. an1 := animate( True(x,t), x=0..1, t=0..2, frames=41, numpoints=65, color=red ): Put animation in a separate window. plotsetup( window ); View animation of true solution. display( an1, insequence=true ); Number of Fourier sine modes to use. N := 3;

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Calculate (and display) first N Fourier sine coefficients. for m from 1 to N do B[m] := evalf( 2*int( sin(m*Pi*x)*f(x), x=0..1 ), 5 ) od;

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Truncated Fourier series solution. Fourier := (x,t) -> sum( B[n]*cos(n*Pi*t)*sin(n*Pi*x), n=1..N );

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Generate (and save) animation plotting data for truncated Fourier solution. an2 := animate( Fourier(x,t), x=0..1, t=0..2, frames=41, numpoints=65, color=blue ): View animation of truncated Fourier solution (use 'inline' or 'default' to see it here in the worksheet). plotsetup( window ); display( an2, insequence=true ); View animations of both solutions together. plotsetup( window ); display( [ an1, an2 ] );