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:

    “Interpreting the dance: young women in white dancing in a ring can only be virgins; old women in black dancing in a ring can only be witches; but middle-aged women in colors, square dancing...?”
    —Mason Cooley (b. 1927)

    “The history of any nation follows an undulatory course. In the trough of the wave we find more or less complete anarchy; but the crest is not more or less complete Utopia, but only, at best, a tolerably humane, partially free and fairly just society that invariably carries within itself the seeds of its own decadence.”
    —Aldous Huxley (1894–1963)