Initial-value problem for a string

A string occupying 0 < x < L has an initial displacement y(0,t) = x(1-x/L) and is released from rest.

(To play the animation again, click on the button on the left.)

Notes

The red curve is the complete solution to the initial-value problem.

The blue curve is the component of the solution that involves the lowest-frequency standing wave.

The initial displacement of the string is close to the lowest standing wave and this remains true at subsequent times.