Gibbs Phenomenon

Following recommendations of the "Decolonizing Rigor in Higher Education" conversation by Mona Abo-Zena and Kerrie Wilkins-Yel, CIT-UMB, Jan 31, 2023

Nuclear Reactor Mishap at Chernobyl, USSR (Wikipedia)

We thank our readers Mason Goad and McGoohan for their help.

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The year is 2023. Alice and Bob are two PhD students in a science department. They are attending a lecture on Fourier analysis. Professor John Smith wants to say:

Rigorously speaking, the L2 convergence does not imply a point-wise convergence. For example, for a sudden jump, your Fourier series will always—no matter how many harmonics—be overshooting the jump by ~10%, by 8.95…% to be precise. The exact formula involves a sine-integral:

\(\frac{1}{\pi } \int_{0}^{\pi}\frac{\sin(t)}{t}dt-\frac{1}{2}\,\,.\)

We have to thoroughly study the derivation; regretfully, it is neither obvious nor intuitive. In similar problems, should they arise in the future, you must conscientiously apply what we learnt; only then the desired accuracy in estimating the true maximal magnitude of the signal can be ensured. Those are the painful but objective facts of Nature.

Unfortunately, Prof. Smith had just attended the Decolonizing Rigor in High Education lecture…

…There he heard, from Mona Abo-Zena and Kerrie Wilkins-Yel, that…

…[w]ithin the context of Western science, rigor presumes truth procedures that are precise, exact, conscientious, accurate, and objective.

And he does not want to appear as a Western colonizer, …

…nor as a White Supremacist.

Hence, Prof. Smith says instead:

Just pump lots of harmonics in, folks. Never worry about series convergence; it doesn’t matter, most of the time.

And so Alice and Bob never learnt about the Gibbs Phenomenon, …

…and they never will. Both will die on June 26, 2035.

The year is now 2035. Location: UMass Boston Laboratory for Safe Nuclear Fusion. And it just got a fresh subcontract from Lawrence Livermore National Laboratory (LLNL) to design the ignition cycle. The goal is a square wave in over-the-ambient pressure. The greater pressure jump the better, but the reaction cell has a break threshold. Tools: a box with harmonics.

Alice [PhD in one science or another, 2024, UMB]:

Let’s just Fourier-transform the square wave, and let’s make it as close to the allowed threshold as we can. And let’s take 125 harmonics; sure thing it will converge by then.

Bob [a PhD in the same science, 2023, UMB]:

Let’s.

And so they did, and LLNL followed their advice. In the experiment that took place a month later the pressure overshot by ~10%: the cell broke, and 2,000+ people died, including Alice and Bob who came to help with the know-how transfer. But, the Decolonize Rigor project lives on.