Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 13B (Methods)

Take positive $y$ vertically downward. For a short string element, the transverse tension resultant is $T{y}_{xx}dx$, while gravity and linear drag contribute $\mu gdx$ and $-2k\mu y_{t}dx$. Newton's second law givesSince $T=\mu c^2$, the damped wave equation isChoosing upward displacement reverses the sign of the gravity term.

With gravity omitted, the initial data contain only the first fixed-end normal mode, so writeThenBecause $k=\pi c/l$, this oscillator is critically damped, andTherefore

The first mode remains the solution from part (b). The applied force excites only the third mode, sowhere, using $\pi c/l=k$,A particular solution isAdding the decaying complementary solution and imposing the initial conditions gives

As $t\to \infty$, all transients decay but the driven third-mode oscillation remains. The string approaches a periodic steady-state response, rather than coming to rest.

With gravity omitted, its mechanical energy isWriting $s=\alpha /(34k^2)$ and letting $t\to \infty$ givesThe limiting energy is periodic because the external force continually supplies the energy dissipated by drag.