Details for an Example

> restart: with(plots):

To see how this vibrating beam is different from a vibrating string, it would be well to compare the two. We displace both these by one arch of the sine function.

> plot(sin(Pi*x),x=0..1);

To keep the two the same, we make c = 1 so that we are comparing solutions for

= and + = 0.

Check that the solution for the string equation with zero boundary conditions and no initial velocity is

> s:=(t,x)->sin(Pi*x)*cos(Pi*t);

> diff(s(t,x),t,t)-diff(s(t,x),x,x);

We graph this solution and also animate graph of the solution.

> plot3d(s(t,x),x=0..1,t=0..2,axes=normal, orientation=[-165,55]);

> animate(s(t,x),x=0..1,t=0..2);

>

Check that the solution for the beam equation with zero boundary conditions and no initial velocity is

> b:=(t,x)->sin(Pi*x)*cos((Pi)^2*t);

> diff(b(t,x),t,t)+diff(b(t,x),x,x,x,x);

We graph this solution and also animate graph of the solution.

> plot3d(b(t,x),x=0..1,t=0..2,axes=normal, orientation=[-165,55]);

> animate(b(t,x),x=0..1,t=0..2);

>

Could you see the difference? The beam vibrated faster.
The string completes one cycle at t = 2, 4, 6, 8, ... . Watch. The following should be three cycles.

> animate(s(t,x),x=0..1,t=0..6);

The beam completes one cycle at t = 2/ , 4/ ,6/ . The following should be three cycles.

> animate(b(t,x),x=0..1,t=0..6/Pi);

>

This suggests that if a beam and a string are both struck, parameters for the two being equal, the beam should vibrate at a high pitch.