June 12, 2005
June 12, 2005
June 15, 2005
10.666.1 - 10.666.11
Gibbs Phenomenon and its Applications in Science and Engineering
Josué Njock Libii
Engineering Department Indiana University-Purdue University Fort Wayne Fort Wayne, Indiana, 46805-1499 Libii@engr.ipfw.edu
Gibbs phenomenon arises in many applications. In this article, the author first discusses a brief history of this phenomenon and several of its applications in science and engineering. Then, using the Fourier series of a square-wave function and computer software in a classroom exercise, he illustrates how Gibbs phenomenon can be used to illustrate to undergraduate students the concept of nonuniform convergence of successive partial sums over the interval from 0 to B.
Gibbs Phenomenon is intimately related to the study of Fourier series. When a periodic function f(x) with a jump discontinuity is represented using a Fourier series, for example, it is observed that calculating values of that function using a truncated series leads to results that oscillate near the discontinuity . As one includes more and more terms into the series, the oscillations persist but they move closer and closer to the discontinuity itself. Indeed, it is found that the series representation yields an overshoot at the jump, a value that is consistently larger in magnitude than that of the actual function at the jump. No matter how many terms one adds to the series, that overshoot does not disappear. Thus, partial sums that approximate f(x) do not approach f(x) uniformly over an interval that contains a point where the function is discontinuous . This behavior, which appears in many practical applications, is known as Gibbs Phenomenon; it is a common example that is used to illustrate how nonuniform convergence can arise . Detailed proofs and demonstrations of Gibbs phenomenon using square waves are found in the literature [2, 3, 22] and on web sites [1, 24, 25, 26].
2. History and applications
This phenomenon is not new. Indeed, an important body of knowledge already exists that relates the history and theory of Gibbs phenomenon to various applications in science and engineering. Such work can be used to enhance the education of engineering students. Gibbs phenomenon was
“Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition Copyright © 2005, American Society for Engineering Education”
Njock-Libii, J. (2005, June), Gibbs Phenomenon In Engineering Systems Paper presented at 2005 Annual Conference, Portland, Oregon. 10.18260/1-2--15299
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