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Chapter 7: Q24P (page 371)

If, in Problem 23, the string is stopped at the center and half of it is plucked, then the function to be expanded in a sine series is shown here. Find the series. Caution: Note thatfor f(x,0) = 0 for l/2<x<l.

Short Answer

Expert verified

The function’s Fourier sine series is

Step by step solution

01

Given Information.

The given function is

02

Meaning of Fourier series

A periodic function f(x) is expanded using the Fourier series formula in terms of an infinite sum of sines and cosines. Any periodic function or periodic signal is decomposed into the sum of a set of simple oscillating functions, mainly sines and cosines. The formula for a function's Fourier series is as follows:

03

Find the Fourier sine series

The function is equal to

Now find the sine series’ coefficients:

Hence, the Fourier sine series is

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Most popular questions from this chapter

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.

sin 2xon(π6,7π6)

Use a computer to produce graphs like Fig. 6.2 showing Fourier series approximations to the functions in Problems 5.1 to 5.3, and 5.7 to 5.11. You might like to set up a computer animation showing the Gibbs phenomenon as the number of terms increases.

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.

x-cos26xon(0,π6)

In Problems 13to 16, find the Fourier cosine transform of the function in the indicated problem, and write f(x)as the Fourier integral [ use equation (12.15)]. Verify that the cosine integral for f(x)is the same as the exponential integral found previously.

15. Problem 9

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.
See all solutions

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