Gibbs Phenomenon - The Square Wave Example

The Square Wave Example

We now illustrate the above Gibbs phenomenon in the case of the square wave described earlier. In this case the period L is, the discontinuity is at zero, and the jump a is equal to . For simplicity let us just deal with the case when N is even (the case of odd N is very similar). Then we have

Substituting, we obtain

as claimed above. Next, we compute

S_N f\left(\frac{2\pi}{2N}\right) = \sin\left(\frac{\pi}{N}\right) + \frac{1}{3} \sin\left(\frac{3\pi}{N}\right) + \cdots + \frac{1}{N-1} \sin\left( \frac{(N-1)\pi}{N} \right).

If we introduce the normalized sinc function, we can rewrite this as

S_N f\left(\frac{2\pi}{2N}\right) = \frac{\pi}{2} \left[ \frac{2}{N} \operatorname{sinc}\left(\frac{1}{N}\right) + \frac{2}{N} \operatorname{sinc}\left(\frac{3}{N}\right) + \cdots + \frac{2}{N} \operatorname{sinc}\left( \frac{(N-1)}{N} \right) \right].

But the expression in square brackets is a numerical integration approximation to the integral (more precisely, it is a midpoint rule approximation with spacing ). Since the sinc function is continuous, this approximation converges to the actual integral as . Thus we have

\begin{align} \lim_{N \to \infty} S_N f\left(\frac{2\pi}{2N}\right) & = \frac{\pi}{2} \int_0^1 \operatorname{sinc}(x)\, dx \\ & = \frac{1}{2} \int_{x=0}^1 \frac{\sin(\pi x)}{\pi x}\, d(\pi x) \\ & = \frac{1}{2} \int_0^\pi \frac{\sin(t)}{t}\ dt \quad = \quad \frac{\pi}{4} + \frac{\pi}{2} \cdot (0.089490\dots), \end{align}

which was what was claimed in the previous section. A similar computation shows

\lim_{N \to \infty} S_N f\left(-\frac{2\pi}{2N}\right) = -\frac{\pi}{2} \int_0^1 \operatorname{sinc}(x)\ dx = -\frac{\pi}{4} - \frac{\pi}{2} \cdot (0.089490\dots).

Read more about this topic:  Gibbs Phenomenon

Famous quotes containing the words square and/or wave:

    “I walked by the Union Square Bar, I was gonna go in. And I saw myself, my reflection in the window. And I thought, “I wonder who that bum is.” And then I saw it was me. Now look at me, I’m a bum. Look at me. Look at you. You’re a bum.”
    —J.P. (James Pinckney)

    ““Justice” was done, and the President of the Immortals, in Æschylean phrase, had ended his sport with Tess. And the d’Urberville knights and dames slept on in their tombs unknowing. The two speechless gazers bent themselves down to the earth, as if in prayer, and remained thus a long time, absolutely motionless: the flag continued to wave silently. As soon as they had strength they arose, joined hands again, and went on.
    The End”
    —Thomas Hardy (1840–1928)