 Straight rotating string with one fixed endpoint

A Nambu–Goto action solution with $X^0=t$ is $\mathbf{X}=L\sin (\sigma /L)(\cos (t/L),\sin (t/L))$, for $0\le \sigma \le \pi L/2$. It solves the wave equation, has a fixed endpoint at the origin, and obeys the free-end string boundary condition at the other end. Its proper length is $L$ and its free endpoint has angular speed $1/L$ and spatial speed one.