Chapter 7: Fourier Series and Transforms

Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance from the origin is the given function.

$\mathit{s}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathit{\pi }\mathit{t}\mathbf{-}\mathbf{8}\mathbf{)}$

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.

For each of the following combinations of a fundamental musical tone and some of its overtones, make a computer plot of individual harmonics (all on the same axes) and then a plot of the sum. Note that the sum has the period of the fundamental.

$\mathbf{sin\pi }\mathbf{}\mathbf{t}\mathbf{+}\mathbf{sin}\mathbf{2}\mathbf{\pi t}\mathbf{+}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{3}\mathbf{}\mathbf{\pi t}$

The diagram shows a “relaxation” oscillator. The chargeqon the capacitor builds up until the neon tube fires and discharges the capacitor (we assume instantaneously). Then the cycle repeats itself over and over.

(a) The charge q on the capacitor satisfies the differential equation

, here R is the Resistance, C is the capacitance and Vis the

Constant d-c voltage, as shown in the diagram. Show that if q=0 when

t=0 then at any later time t (during one cycle, that is, before the neon

Tube fires),

(b) Suppose the neon tube fires at. Sketch q as a function of t for

several cycles.

(b) Expand the periodic q in part (b) in an appropriate Fourier series.

In Problemsto, the sketches show several practical examples of electrical signals (voltages or currents). In each case we want to know the harmonic content of the signal, that is, what frequencies it contains and in what proportions. To find this, expand each function in an appropriate Fourier series. Assume in each case that the part of the graph shown is repeated sixty times per second.

. Output of a simple d-c generator; the shape of the curve is the absolute value of a sine function. Let the maximum voltage be 100V.

Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance from the origin is the given function.

$\mathit{s}\mathbf{=}\mathbf{5}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathit{t}\mathbf{-}\mathit{\pi }\mathbf{)}$

Find the exponential Fourier transform of the given f(x)and write f(x) as a Fourier integral [that is, find $\mathbf{\text{g}}\mathbf{\left(}\mathbf{\text{α}}\mathbf{\right)}$in equation (12.2) and substitute your result into the first integral in equation (12.2)].

role="math" localid="1664339168986" $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{l}\begin{array}{ll}\mathbf{1}\mathbf{,}& \mathbf{\pi }\mathbf{/}\mathbf{2}\mathbf{<}\mathbf{\left|}\mathbf{x}\mathbf{\right|}\mathbf{<}\mathbf{\pi }\end{array}\\ \begin{array}{ll}\mathbf{0}\mathbf{,}& \mathbf{\text{otherwise}}\end{array}\end{array}$

When a current Iflows through a resistance, the heat energydissipated per secondis the average value of$\mathit{R}{\mathit{I}}^{\mathbf{2}}$. Let a periodic (not sinusoidal) current I(t) be expanded in a Fourier series$\mathbf{I}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{\sum }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}{\mathbf{c}}_{\mathbf{n}}{\mathbf{e}}^{\mathbf{120}\mathbf{i}\mathbf{n}\mathbf{\pi }\mathbf{t}}$.Give a physical meaning to Parseval’s theorem for this problem.

For each of the following combinations of a fundamental musical tone and some of its overtones, make a computer plot of individual harmonics (all on the same axes) and then a plot of the sum. Note that the sum has the period of the fundamental.

$\mathit{c}\mathit{o}\mathit{s}\mathbf{2}\mathit{\pi }\mathit{t}\mathbf{+}\mathit{c}\mathit{o}\mathit{s}\mathbf{4}\mathit{\pi }\mathit{t}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathit{c}\mathit{o}\mathit{s}\mathbf{6}\mathit{\pi }\mathit{t}$

In Problems 3 to 12, find the average value of the function on the given interval. Use equation (4.8) if it applies. If an average value is zero, you may be able to decide this from a quick sketch which shows you that the areas above and below the x axis are the same.

$\mathbf{1}\mathbf{-}{\mathit{e}}^{\mathbf{-}\mathbf{x}}\mathit{o}\mathit{n}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}$