Chapter 13.4, Problem 5P

A string of length $l$ is initially stretched straight; its ends are fixed for all t .At time $t=0,$ its points are given the velocity $V(x)={(\partial y/\partial t)}_{t=0}$ as indicated in the diagram (for example, by hitting the string). Determine the shape of the string at time t, that is,find the displacement $y$ as a function of $x$ and $t$ in the form of a series similar to (4.9). Warning: What basis functions do you need here? Answer: $y=\frac{8hl}{{\pi }^{3}v}\left(\sin \frac{\pi x}{l}\sin \frac{\pi vt}{l}-\frac{1}{3^3}\sin \frac{3\pi x}{l}\sin \frac{3\pi vt}{l}+\frac{1}{5^3}\sin \frac{5\pi x}{l}\sin \frac{5\pi vt}{l}-\cdots \right)$.