Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 18 / c / ii

Constant translation speed gives $\boxed{\ell (t)=vt}$, since the initial displacement is zero. Substituting into the Fourier series givesThe periodic ramp moves in the positive $x$ direction for $v>0$. At the moving jumps, this series gives the average of the two one-sided values.