Chapter 7: Fourier Series and Transforms

In Problems 17 to 22 you are given f(x) on an interval, say 0 < x < b. Sketch several periods of the even function f_cof period 2b, the odd function f_sof period 2b, and the function f_pof period b, each of which equals f(x)on 0 < x < b . Expand each of the three functions in an appropriate Fourier series.

In Problems 17to 20, find the Fourier sine transform of the function in the indicated problem, and write f(x)as the Fourier integral [ use equation (12.14)]. Verify that the sine integral for f(x)is the same as the exponential integral found previously.

17.Problem 3

In Problems 17 to 22 you are given f(x) on an interval, say 0 < x < b. Sketch several periods of the even function f_cof period 2b, the odd function f_sof period 2b, and the function f_pof period b, each of which equals f(x)on 0 < x < b . Expand each of the three functions in an appropriate Fourier series.

In Problems 17to 20, find the Fourier sine transform of the function in the indicated problem, and write f(x)as a Fourier integral [use equation (12.14)]. Verify that the sine integral for f(x)is the same as the exponential integral found previously.

Problem 6.

Each of the following functions is given over one period. Sketch several periods of the corresponding periodic function and expand it in an appropriate Fourier series.

f(x) = x² , 0 < x < 10

Each of the following functions is given over one period. Sketch several periods of the corresponding periodic function and expand it in an appropriate Fourier series.

In Problems 17to 20,find the Fourier sine transform of the function in the indicated problem, and write f(x)as a Fourier integral [use equation (12.14)]. Verify that the sine integral for f(x)is the same as the exponential integral found previously.

Problem 10.

In Problems 17 to 22 you are given f(x) on an interval, say 0 < x < b. Sketch several periods of the even function f_cof period 2b, the odd function f_sof period 2b, and the function f_pof period b, each of which equals f(x)on 0 < x < b . Expand each of the three functions in an appropriate Fourier series.

19.

The displacement (from equilibrium) of a particle executing simple harmonic motion may be either$\mathit{y}\mathbf{=}\mathit{A}\mathbf{sin}\mathit{\omega }\mathit{t}$or$\mathit{y}\mathbf{=}\mathit{A}\mathbf{sin}\mathbf{(}\mathbf{\omega }\mathbf{t}\mathbf{+}\mathbf{\varphi }\mathbf{)}$depending on our choice of time origin. Show that the average of the kinetic energy of a particle of mass m(over a period of the motion) is the same for the two formulas (as it must be since both describe the same physical motion). Find the average value of the kinetic energy for the$\mathbf{sin}\mathbf{(}\mathbf{\omega }\mathbf{t}\mathbf{+}\mathbf{\varphi }\mathbf{)}$case in two ways:

(a) By selecting the integration limits (as you may by Problem 4.1) so that a change of variable reduces the integral to thecase.

(b) By expanding$\mathbf{sin}\mathbf{(}\mathbf{\omega }\mathbf{t}\mathbf{+}\mathbf{\varphi }\mathbf{)}$by the trigonometric addition formulas and using (5.2) to write the average values.

In problem 1to 3, the graphs sketched represent one period of the excess pressure p(t)in a sound wave. Find the important harmonics and their relative intensities. Use a computer to play individual terms or a sum of several terms of the series.