Normal modes in a loaded string

A light string is under tension T between two fixed points at positions x₀ and x_N₊₁ = x₀+L and N equal masses m are located at regular intervals, i.e., at positions x_s = sa, with a = L/(N+1) and s = 1,2,...,N.

The masses may oscillate in the transverse direction in a plane. The oscillation amplitude is always small.

The simulation will display the normal modes for N = 1,2,...,10.
One may also get the envolvent of amplitudes. (This will be useful to compare with the continuous vibrating 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 equations of motion are

where ω₀ = (T/ma)^1/2.
Check that periodic solutions in the form y_s = C_s e^iωt satisfy the equations of motion when the frequency ω equals one of the normal frequencies

and the amplitudes are given by
Use the simulation to check your results.
In particular, select A to see the sinusoidal envolvent of the amplitudes for each mode.
Which is the fastest mode? And the slowest one?
In which case(s) can a mass remain at rest?
Write the expression of the general oscillation with N masses.

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.