Motion of a clamped/clamped string after one clamp is released:

NOTE added 31 January 2013: This animation was devised in 2005 to illustrate a principle developed in Oren Elrad's thesis (2006) as a tentative solution of "The Feynman Spaghetti Problem." Many websites (which include a few videos) discuss the problem. The original notebook was written in Mathematica v6.

The string in this 120-frame film loop executes one vibration in its fundamental mode, whereupon the right clamp is released. Release causes the period to double: 40 frames are devoted to the oscillation that occurs just prior to release, 80 frames to the first complete oscillation after release.

Here I define the wave functions prior to and after release

ϕ[x_,t_]:=(1/2)(Cos[π t]+Sin[π t])Sin[π x]
Φ[x_,t_,a_,b_,p_]:=(∑_(k=0)^p)(a(8/((4-(((2k+1))^2))π))Sin[(((2k+1)π)/2)]Cos[(((2k+1))/2)π t]+
b(2/(2k+1))(8/((4-(((2k+1))^2))π))Sin[(((2k+1)π)/2)]Sin[(((2k+1))/2)π t])Sin[(((2k+1)π)/2)x]

NOTE: I seem to have lost the calculations that generated those constructions. The underlying principles are presented in Chapter 6 of the Sophomore Physics notes.

...and here I construct the film:

Table[{If[n<40,{Plot[ϕ[x,((n-40)/20)],{x,0,1}, PlotRange->{-1.5,1.5}, Ticks->{{1},{-1,1}}]},{Plot[Φ[x,((n-40)/20),1/2,1/2,15],{x,0,1}, PlotRange->{-1.5,1.5},Ticks->{{1},{-1,1}}]}]},{n,0,120}];

NOTE: The conversion from v6 to v7 has resulted in a command that does not stop running until aborted (Control Period).

Click on the image to animate.

Note that the pulse reverses sign when it bounces off the clamped end of the string, but not when it bounces off the unclamped end. It is just such a pulse that snaps "Feynman's spaghetti."

NOTE: The persistent zero-derivative at the unclamped end is forced by a momentum density consideration.

NOTE: Here is an alternative command that is free from the "runs until stopped" problem, but (even when slowed down) it results for some reason in a much jerkier movie:

Manipulate[Which[n<40,Plot[ϕ[x,((n-40)/20)],{x,0,1}, PlotRange->{-1.5,1.5}, Ticks->{{1},{-1,1}}],n>39,Plot[Φ[x,((n-40)/20),1/2,1/2,15],{x,0,1}, PlotRange->{-1.5,1.5},Ticks->{{1},{-1,1}}]],{n,0,120,1}]