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Chapter 9: Problem 14

The symmetric square wave $$ f(x)=\left\\{\begin{array}{ll} \text { 1, } & |x|<\frac{\pi}{2} \\ -1, & \frac{\pi}{2}<|x|<\pi \end{array}\right. $$ has a Fourier expansion $$ f(x)=\frac{4}{\pi} \sum_{n=0}^{\infty}(-1)^{n} \frac{\cos (2 n+1) x}{2 n+1} $$ Evaluate this series for \(x=0(\pi / 18) \pi / 2\) using the first (a) 10 terms, (b) 100 terms of the series. Note. As in Exercise 9.2.13, the Gibbs phenomenon appears at the discontinuity. This means that a Fourier series is not suitable for precise numerical work in the vicinity of a discontinuity.

Short Answer

Expert verified
Evaluate the Fourier series for \( x = 0, \pi/18, \pi/2 \) using 10 and 100 terms, observing Gibbs phenomenon near discontinuities.

Step by step solution

01

Understand the Fourier Series Representation

The given Fourier series for the square wave function is \( f(x)=\frac{4}{\pi} \sum_{n=0}^{\infty}(-1)^{n} \frac{\cos ((2 n+1) x)}{2 n+1} \). This series will be evaluated for different values of \( x \) using a specific number of terms \( N \).
02

Setup x Evaluations

For this problem, we'll evaluate the series at \( x = 0 \), \( x = \pi/18 \), and \( x = \pi/2 \). Each value will be substituted into the Fourier series to calculate the series sum.
03

Calculate Sum for 10 Terms

For each \( x \) value, calculate the sum of the first 10 terms of the series. This involves substituting appropriate values of \( n \) from 0 to 9 in the series formula and adding them together: \[ S_{10}(x) = \frac{4}{\pi} \sum_{n=0}^{9}(-1)^{n} \frac{\cos ((2 n+1) x)}{2 n+1} \].
04

Calculate Sum for 100 Terms

Similarly, for each \( x \) value, calculate the sum of the first 100 terms of the series. Use the formula: \[ S_{100}(x) = \frac{4}{\pi} \sum_{n=0}^{99}(-1)^{n} \frac{\cos ((2 n+1) x)}{2 n+1} \].
05

Evaluate at \( x = 0 \)

Substitute \( x = 0 \) into both the 10-term and 100-term expressions and compute the sums sequentially. Since \( \cos(k \cdot 0) = 1 \) for any integer \( k \), each term simplifies accordingly.
06

Evaluate at \( x = \pi/18 \)

Substitute \( x = \pi/18 \) in the series for both 10 terms and 100 terms sums. Calculate the \( \cos((2n+1) \, \pi/18) \) values for each \( n \) and sum them.
07

Evaluate at \( x = \pi/2 \)

Substitute \( x = \pi/2 \) into the series expression for both 10 and 100 terms. Note that this evaluation might show the Gibbs phenomenon as \( x = \pi/2 \) is near a discontinuity.
08

Compare Results and Discuss Gibbs Phenomenon

Compare the computed results at each \( x \) for both 10 and 100 terms. Notice the difference, specifically at \( x = \pi/2 \), where the Gibbs phenomenon could affect accuracy due to proximity to a discontinuity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Symmetric Square Wave
The symmetric square wave is a periodic function that alternates between two constant values, typically +1 and -1. It is defined within a specific range, abruptly changing value at certain points. This alternation makes it a non-smooth function, with known jumps at specified locations. In mathematical terms, it can be expressed using a formula such as:
\[ f(x)=\left\{\begin{array}{ll}1, & |x|<\frac{\pi}{2} \-1, & \frac{\pi}{2}<|x|<\pi\end{array}\right. \]
  • The square wave is known for its clear and sudden transitions between values, making it an interesting candidate for analysis using Fourier series.
  • Despite its simple definition, representing it precisely using Fourier series poses challenges due to its inherent discontinuities.
  • Understanding the behavior of a square wave under Fourier expansion is critical for fields like signal processing.
The representation of such a wave using Fourier series showcases how infinite sums of sine and cosine functions can approximate piecewise functions.
Gibbs Phenomenon
The Gibbs phenomenon is a critical concept in the study of Fourier series, particularly when dealing with functions that contain points of discontinuity like a symmetric square wave. When approximating such functions using Fourier series, the series doesn't just jump directly to the new value at a point of discontinuity.
Instead, there are noticeable overshoots and oscillations near these points.
  • This overshoot does not decrease as more terms are added, stabilizing at roughly 9% of the jump irrespective of the number of terms in the series.
  • As more terms are included in the series, the area impacted by the overshoot narrows, but the magnitude of the overshoot remains the same.
  • This phenomenon is a direct result of the series attempting to align itself with the rapid changes or jumps in the function.
Understanding the Gibbs phenomenon is crucial when using Fourier series for numerical evaluations, as it affects the accuracy of approximations close to discontinuities.
Discontinuity
Discontinuity in mathematical functions refers to points or intervals where a function is not continuous. In simpler terms, a discontinuity is where a graph of a function has a distinct "break" or "jump."
  • For the symmetric square wave, discontinuities occur where the function suddenly jumps from one value to another.
  • These points, like at \( x = \frac{\pi}{2} \), present challenges during Fourier series approximation.
  • Discontinuities are inherent in piecewise functions, which are defined by multiple expressions.
When a Fourier series is used to approximate these functions, the Gibbs phenomenon surfaces near these discontinuities, highlighting a fundamental limitation of this method. Understanding the nature of discontinuities can aid in effectively applying and interpreting Fourier series approximations.
Numerical Evaluation
Numerical evaluation of a Fourier series involves calculating the sum of its terms with specific values substituted for variables like \( x \). This process helps mathematically approximate functions like the symmetric square wave.
  • The accuracy of the approximation depends on the number of terms evaluated in the series.
  • Using more terms generally gives a more accurate approximation, but comes with increased computational work.
  • The exercise here evaluates the Fourier series at specific \( x \) values using both 10 and 100 terms, demonstrating convergence properties and accuracy changes.
Additionally, when performing numerical evaluations near discontinuities, it's crucial to recognize the influence of the Gibbs phenomenon, as it can lead to inaccuracies in the results. This understanding can guide better placements of numerical evaluations away from critical points, ensuring more reliable outputs.

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Most popular questions from this chapter

\(A\) and \(B\) are noncommuting quantum mechanical operators: $$ A B-B A=i C $$ Show that \(C\) is Hermitian. Assume that appropriate boundary conditions are satisfied.

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