The Fourier series representation of a square wave is shown in the figure below. The fluctuations seen near \(x = \pm 1\) are named after which one of the following scientists?
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These overshoot ripples near a jump discontinuity never shrink in height as more Fourier terms are added, only in width. This named effect is a classic property of truncated Fourier series.
The plot shows a truncated Fourier series trying to reproduce a square wave, which has sharp jump discontinuities at \(x = \pm 1\). Near each jump, the reconstructed curve overshoots the true value, dips, and rings before settling down. This ringing pattern is a well known named effect in Fourier analysis, and matching it to the right name is what this question tests.
Cauchy: Augustin-Louis Cauchy's name is attached to results in complex analysis and to the Cauchy sequence and Cauchy integral theorem, not to the overshoot near a jump in a Fourier series. This does not match.
Fourier: Joseph Fourier gave his name to the series itself, the representation of a periodic function as a sum of sines and cosines. The series is called Fourier series throughout, but the specific ringing artifact near a discontinuity is credited to a different mathematician, not to Fourier.
Gibbs: Josiah Willard Gibbs analyzed exactly this behavior. When a Fourier series for a function with a jump discontinuity is truncated to a finite number of terms, the partial sum overshoots the jump by a fixed fraction, about 9 percent of the jump size, and this overshoot does not vanish even as more terms are added, it only squeezes closer to the discontinuity. This is precisely the ringing seen near \(x=\pm1\) in the figure, and it is called the Gibbs phenomenon.
Laplace: Pierre-Simon Laplace is known for the Laplace transform and Laplace's equation, both different tools used elsewhere in engineering mathematics, and unrelated to this specific Fourier series overshoot behavior.
The correct option is Gibbs, since the fixed size overshoot and ringing near a jump discontinuity of a truncated Fourier series is specifically named the Gibbs phenomenon.
Let's sum up:
A square wave has a jump discontinuity at \(x=\pm1\), and no finite Fourier sum can reproduce a jump exactly.
Near the jump, the partial sum overshoots by roughly 9 percent of the jump height, no matter how many terms are used.
This persistent overshoot and ringing pattern is called the Gibbs phenomenon, named after J. Willard Gibbs.
So the fluctuations near \(x=\pm1\) are named after Gibbs.
