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Chapter 1: Problem 4

It is desired to express the function \(y(x)=l x-x^{2}\) in the interval \(0 \leq x \leq l / 2\) by a Fourier series. Investigate how to piece it out so that it becomes a periodic function, of period \(2 l\), so that the Fourier series converges as rapidly as possible. Obtain the Fourier series and plot curves showing the sum of one, two, and three terms of the series, as well as the function \(y(x)\) that it represents.

Short Answer

Expert verified
Question: Express the function \(y(x) = l x - x^2\) on the interval \(0 \leq x \leq l/2\) as a Fourier series and plot the sum of one, two, and three terms of the series along with the original function. Answer: The Fourier series for the given function can be expressed as: $$ y_p(x) \approx \sum_{n=1}^{\infty}b_n \sin{\frac{n\pi x}{l}} $$ where the Fourier coefficients, \(b_n\), are found by calculating the following integrals: $$ b_n = \frac{2}{l}\left(\int_0^{l/2}(lx-x^2) \sin{\frac{n\pi x}{l}} dx + \int_{l/2}^l (l(2l-x) - (2l-x)^2) \sin{\frac{n\pi x}{l}} dx\right) $$ Plot the original function, \(y(x) = lx - x^2\), and the sums of one, two, and three terms of the Fourier series, \(y_1(x)\), \(y_2(x)\), and \(y_3(x)\), to visualize the convergence within the interval \(0 \leq x \leq l/2\).

Step by step solution

01

Make the function periodic

First, we need to extend the function \(y(x) = l x - x^2\) on the interval \(0 \leq x \leq l/2\) to create a periodic function with a period of \(2l\). To do this, we'll piece together the function, symmetrizing it about the midpoint and reflecting it with respect to the y-axis. The extended function can be defined as follows: $$ y_{p}(x) = \begin{cases} l x - x^2, &0 \leq x \leq l/2 \\ l(2l-x) - (2l-x)^2, &l/2 \leq x \leq l \\ -y_{p}(x-2l), &l \leq x \leq 3l \\ -y_{p}(x-4l), &3l \leq x \leq 5l\\ \cdots \end{cases} $$
02

Calculate the Fourier series

Next, we will calculate the Fourier series for the extended function \(y_p(x)\). Since the function \(y_p(x)\) is odd, we will only consider the sine terms in the Fourier series: $$ y_p(x) \approx \sum_{n=1}^{\infty}b_n \sin{\frac{n\pi x}{l}} $$ Using the sine terms of the Fourier series formula: $$ b_n = \frac{2}{l} \int_0^l y_p(x) \sin{\frac{n\pi x}{l}} dx $$ We can split the integration in two parts, as we have defined two different pieces for \(y_p(x)\): $$ b_n = \frac{2}{l}\left(\int_0^{l/2}(lx-x^2) \sin{\frac{n\pi x}{l}} dx + \int_{l/2}^l (l(2l-x) - (2l-x)^2) \sin{\frac{n\pi x}{l}} dx\right) $$ Calculate the above integrals to find the Fourier coefficients \(b_n\).
03

Plot the sum of one, two, and three terms

After finding the Fourier coefficients, a Fourier series with one, two, and three terms will be: $$ y_1(x) \approx b_1\sin{\frac{\pi x}{l}} $$ $$ y_2(x) \approx b_1\sin{\frac{\pi x}{l}} + b_2\sin{\frac{2\pi x}{l}} $$ $$ y_3(x) \approx b_1\sin{\frac{\pi x}{l}} + b_2\sin{\frac{2\pi x}{l}} + b_3\sin{\frac{3\pi x}{l}} $$ Plot the following curves together to visualize the convergence: 1. the original function \(y(x) = lx-x^2\) 2. the sum of the first term, \(y_1(x)\) 3. the sum of the first two terms, \(y_2(x)\) 4. the sum of the first three terms, \(y_3(x)\) Observe the convergence of the Fourier series to the original function within the interval \(0 \leq x \leq l/2\).

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Most popular questions from this chapter

The two waves \(\eta_{1}=A_{1} \cos (\kappa x-\omega t)\) and \(\eta_{2}=A_{2} \cos (\kappa x+\omega t)\) travel together on a stretched string. Show how the wave amplitude ratio \(A_{1} / A_{2}\) can be found from a measurement of the so-called standing-wave ratio \(S\) of the resulting wave pattern, which is defined as the ratio of the maximum amplitude to the minimum amplitude of the envelope of this pattern. Note that \(S\) and the position and amplitude of the minima (or maxima) wave disturbance uniquely determine the two traveling waves that give rise to the standing-wave pattern.

Two transverse symmetrical sawtooth waves, each having the form at \(t=0\) shown in the figure are traveling in opposite directions on a stretched string. Investigate the resulting disturbance and plot the wave pattern at several times in the time interval \(0 \leq t \leq l / c\). Does the pattern in the range \(0 \leq x \leq l\) correspond to Fig. \(1.6 .2\), which is a plot of \(\eta(x, l)\) as given by (1.6.9)?

Show that the Fourier series (1.7.1) can be expressed in the complex form $$ f(\theta)=\sum_{n=-\infty}^{+\infty} A_{n} e^{i n \theta} $$ where the complex Fourier coefficients are given by $$ \breve{A}_{\mathrm{n}}=\frac{1}{2 \mathrm{x}} \int_{-\pi}^{\pi} f(\theta) e^{-i \text { in } \theta} d \theta $$ Here, the entire series \((1.7 .11)\), not just its real part, represents \(f(\theta)\). If \(f(\theta)\) is real, this implies that the imaginary part of the series sums to zero. Relate the \(A_{n}\) to the coefficients \(a_{0}, a_{n}, b_{n}\) given by (1.7.2). We make considerable use of Fourier series in the complex form in later chapters.

Investigate the motion of a string having transverse waves on it specified by the equations \(\eta=A \sin (\kappa x-\omega t)\) and \(\zeta=A \cos (x x-\omega t)\). If the string, instead of being infinitely long, is attached to a wall, so that reflections occur, find what sort of standing-wave pattern exists. Does the motion have a connection with rope jumping \(^{2}\) \(\dagger\) For an analysis of the nonlinear vibration of a string segment with fixed ends, see G. F. Carrier, Quart. Appl. Math., 3: 157 (1945).

Show how the solution \((1.5 .5)\) of the wave equation can be transformed into a general traveling-wave solution.
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