Week 9

String Theory

In class, we made the transition from a finite number of coupled oscillators to the continuum limit (our system of study was a string with N masses on it, and the continuum limit involved taking N to infinity while holding the length of the system L = (N+1) l constant and the total mass of the system M=(N*m) fixed).  This led to our discovery of the wave equation.  The normal mode solutions to the wave equation when the string is fixed at x = 0 and x = L are given by:

L=1;v=1;

Taking L = 1m and v = 1m/s-

y[x_,t_,n_,CC_,α_]=CC Sin[n π x/L] Cos[n π v/L t + α]

Animate[Plot[{y[x,t,1,1,0],y[x,t,2,1,0],y[x,t,3,1,0],y[x,t,4,1,0]},{x,0,L},PlotRange->{-1,1}],{t,0,10π},AnimationRate->.01]

Animate[Plot[y[x,t,1,1,0]+y[x,t,2,1,0]+y[x,t,3,1,0]+y[x,t,4,1,0],{x,0,L},PlotRange->{-3,3}],{t,0,10π},AnimationRate->.01,RefreshRate->100]

Animate[Plot[Sum[ y[x,t,n,1,0],{n,0,100}],{x,0,L},PlotRange->{-10,10},PlotPoints->100],{t,0,10π},AnimationRate->.01]