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

Apart from its endpoint convention, the ramp is odd, so its constant and cosine Fourier coefficients vanish. For the sine coefficient, evenness of the integrand and integration by parts giveHenceThis is the Fourier series of a periodic sawtooth. At the periodic jump, the series sums to zero rather than the assigned endpoint value $K/2$; changing a value at one endpoint does not change the coefficients.

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.

Not as ordinary convergent Fourier series, for a nontrivial moving ramp. Formal spatial differentiation would giveand formal time differentiation multiplies that expression by $-v$. The summands have nondecaying amplitude and do not tend to zero, so the ordinary series cannot converge. The periodic extension has jumps, violating the regularity needed for Termwise differentiation of a Fourier series.

Away from the moving jumps, the classical derivatives exist and are $g_x=K/L$ and $g_t=-vK/L$. A distributional derivative also includes the Dirac delta distribution contributions at the jumps, so distributional differentiation is possible, but it is a different interpretation. If $v=0$, the time derivative is trivially zero; if $K=0$, both derivatives are zero.