Forced vibrations of a string

The ends of a string of length L are oscillated according to

y(0,t) = 0 and y(L,t) = a cos ωt

so that the displacement

y(x,t) = a sin(ωx/c) cos ωt / sin(ωL/c)

Resonance occurs for ω = n π c / L; these are the frequencies of the standing wave solutions for this string.

The frequency ω = 0.95 π c / L is close to the first resonant frequency and there is a large oscillation with "shape" close to that of the first standing wave.

The frequency ω = 1.5 π c / L is not close to any resonant frequency and the motion of the string is not large.

The frequency ω = 1.95 π c / L is close to the second resonant frequency and there is a large oscillation with "shape" close to that of the second standing wave.