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?
Show Hint
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.
Step 1: Understanding the Question:
We are shown a finite, truncated Fourier series trying to approximate a square wave, which jumps sharply at $x=\pm1$. Near those jump points the curve does not settle smoothly, it overshoots above the top level and undershoots below the bottom level in a rippling pattern, and we are asked which scientist's name is attached to that specific ripple.
Step 2: Key Formula or Approach:
A square wave has a sharp jump, so it is not smooth at $x=\pm1$. Trying to write it as a finite sum of sines and cosines can only approximate a jump, never reproduce it exactly. Near the jump, this partial sum always overshoots the true jump size by a fixed amount, close to 9 percent of the jump height, and this overshoot amount stays roughly the same size even as more terms are added to the sum. The overshoot region just gets narrower and squeezes in closer to the jump, it never disappears.
Step 3: Detailed Explanation:
Go through the choices one at a time. Cauchy's name belongs to complex analysis results like the Cauchy integral formula, not to this ripple. The word Fourier already names the series being used to build the approximation, so it cannot also be the special name for the leftover ripple error, that would be redundant. Laplace's name belongs to the Laplace transform and Laplace's equation, tools used for solving differential equations, again unrelated to this ripple near a jump. Gibbs is the correct match: J. Willard Gibbs is credited with explaining why this fixed size overshoot persists near a jump discontinuity of a truncated Fourier series, and the effect carries his name, the Gibbs phenomenon.
Step 4: Final Answer:
The named ripple seen near $x=\pm1$ in the figure is the Gibbs phenomenon, so the correct option is Gibbs.
\[ \boxed{\text{Gibbs phenomenon}} \]
