Chapter 7: Fourier Series and Transforms

The velocity of sound in sea water is about $\mathbf{1530}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{sec}$. Write an equation for a sinusoidal sound wave in the ocean, of amplitude 1 and frequency$\mathbf{1000}\mathbf{}\mathbf{hertz}$ .

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\{\begin{array}{l}\begin{array}{ll}\sin x,& 0<x<\pi \end{array}\\ \begin{array}{ll}0,& \mathrm{otherwise}\end{array}\end{array}$

Hint: write $\mathbf{sin}\mathit{x}$in complex exponential form.

(b) Show that your result can be written as

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{\pi }}\underset{0}{\overset{\infty }{\int }}\frac{\cos \alpha x+\cos \alpha (x-\pi )}{1-{\alpha }^2}d\alpha$.

Suppose that f(x)and its derivative f'(x)are both expanded in Fourier series on. Call the coefficients in the f(x)series a_nand b_nand the coefficients in the f'(x)series a'_nand a'_n. Write the integral for a_n[equation (5.9)] and integrate it by parts to get an integral of f'(x)sinnx. Recognize this integral in terms of[equation (5.10) for f'(x)] and so show that b'_n = -na_n. (In the integration by parts, the integrated term is zero becausesince fis continuous— sketch several periods.). Find a similar relation for a'_nand b'_n. Now show that this is the result you get by differentiating the f(x)series term by term. Thus you have shown that the Fourier series for f'(x)is correctly given by differentiating thef(x)series term by term (assuming that f'(x)is expandable in a Fourier series).

Write an equation for a sinusoidal radio wave of amplitude 10 and frequency$\mathbf{600}\mathbf{}\mathbf{kilohertz}$. Hint: The velocity of a radio wave is the velocity of light,$\mathbf{c}\mathbf{=}\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{sec}$

Represent each of the following functions (a) by a Fourier cosine integral, (b) by a Fourier sine integral. Hint: See the discussion just before theParseval’s theorem.

In Problem 26 and 27, find the indicated Fourier series. Then differentiate your result repeatedly (both the function and the series) until you get a discontinuous function. Use a computer to plot $\mathit{f}\left(x\right)$and the derivative functions. For each graph, plot on the same axes one or more terms of the corresponding Fourier series. Note the number of terms needed for a good fit (see comment at the end of the section).

$\mathbf{26}\mathbf{.}\mathit{f}\left(x\right)\mathbf{=}\{\begin{array}{ll}3x^2+2x^3,& -1<x<0\\ 3x^2-2x^3,& 0<x<1\end{array}$

In Problem and , find the indicated Fourier series. Then differentiate your result repeatedly (both the function and the series) until you get a discontinuous function. Use a computer to plot f(x)and the derivative functions. For each graph, plot on the same axes one or more terms of the corresponding Fourier series. Note the number of terms needed for a good fit (see comment at the end of the section).

Represent each of the following functions (a) by a Fourier cosine integral, (b) by a Fourier sine integral. Hint: See the discussion just before theParseval’s theorem

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{l}\begin{array}{ll}\mathbf{1}\mathbf{,}& \mathbf{0}\mathbf{<}\mathbf{x}\mathbf{<}\frac{\mathbf{\pi }}{\mathbf{2}}\end{array}\\ \begin{array}{ll}\mathbf{0}\mathbf{,}& \mathbf{x}\mathbf{>}\frac{\mathbf{\pi }}{\mathbf{2}}\end{array}\end{array}$

Represent each of the following functions (a) by a Fourier cosine integral; (b) by a Fourier sine integral. Hint: See the discussion just before Parseval’s theorem.

28.$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\{\begin{array}{l}1,2<x<4\\ 0,0<x<2,x>4\end{array}$

Represent each of the following functions (a) by a Fourier cosine integral; (b) by a Fourier sine integral. Hint: See the discussion just before Parseval’s theorem.

29.$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\{\begin{array}{l}-1,0<x<2\\ 1,2<x<3\\ 0,x>3\end{array}$