Standing waves in the vibrating string

A string of mass m and length L is under tension T between its fixed end points. It can vibrate in a plane in the direction perpendicular to its rest position:

The simulation will display the first few normal modes.
One may also get the amplitudes. (This will be useful to compare with the discrete loaded string in the corresponding simulation.)
To get information on an element, put over it the mouse pointer to see the corresponding tooltip.

Activities

Show that the equation of motion is the wave equation

where the phase velocity is v = (LT/m)^1/2.
Check that periodic solutions in the form y(t,x) = A(x) e^iωt satisfy the wave equation and the boundary conditions y(t,0) = y(t,L) = 0 when the frequency ω equals one of the normal frequencies (harmonics) ω = nπv/L and the amplitude is given by A(x) = C sin(nπx/L).
Use the simulation to check your results.
In particular, select A to see the amplitude of each mode.
Which is the fastest mode? And the slowest one?
How many points remain at rest (are nodes) in each mode?

This is an English translation of the Basque original for a course on mechanics, oscillations and waves.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.