Solution (source code)

= Solution

Most of the <mean-square signal power> lies at low <frequencies>. Each nonzero coefficient has magnitude $2/(\pi|r|)\,\mathrm V$, so its power contribution decreases as $r^{-2}$. The pair at the fundamental frequencies $\pm0.25\,\mathrm{Hz}$ alone contains
$$
\boxed{\frac8{\pi^2}\simeq81.1\%}
$$
of the total. The first three positive odd harmonics, together with their negative-frequency partners, contain about $93.3\%$. The <harmonic power of a square wave> still has a nonzero tail: higher harmonics are needed to reproduce sharp edges even though their combined power is small.

\Image[/past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2024/paper-3-q7-square-wave-power.png]
{title=Odd Fourier harmonics reconstruct a unit square wave and concentrate most power in the fundamental}
{description=Original analytic calculations. The reconstruction and paired-harmonic powers use the derived exact coefficients, rather than the supplied noisy spectral illustration. The horizontal axis is time in seconds; voltage and mean-square voltage units are retained.}