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Chapter 7: Q7P (page 347)

Use a trigonometry formula to write the two terms as a single harmonic. Find the period and amplitude. Compare computer plots of your result and the given problem.

cosπx-cosπ(x-1/2)

Short Answer

Expert verified

The period and amplitude are T = 2 and A=2.

Step by step solution

01

Given information

A fundamental musical tone and some of its overtones of harmonics are:

cosπx-cosπ(x-1/2)

02

Formula of trigonometry

Trigonometry formula is[sina+sinb=2sina+b2cosa-b2].

03

Write the two terms as a single harmonic

The given harmonic term iscosπx-cosπ(x-1/2).

=-2sinπx+πx-π/22sinπx+πx-π/22=-2sinπx-π4sinπ4=-2sinπx-14

Thus,the period and amplitude are T = 2 and A=2.

04

The computer plots of the start and end functions

The plot of start of the function is shown below

And, plot of function is shown below.end of the

Thus, the period and amplitude are T = 2 and A=2.

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Most popular questions from this chapter

Expand the same functions as in Problems 5.1 to 5.11 in Fourier series of complex exponentials einx on the interval (-π,π)and verify in each case that the answer is equivalent to the one found in Section 5.

Given f(x)={x,0<x<1-2,1<x<2

a) Sketch at least three periods of the graph of the function represented by the sine series for f(x). Without finding any series, answer thefollowing question:

b) To what value does the sine series in (a) converge at x=1? At x=2? At x=0? At x=-1?

c)If the given function is continued with the period 2and then is represented by a complex exponential series ∑n=-∞∞Cneinπx, what is the value of ∑n=−∞∞|cn|2?

(a) Represent as an exponential Fourier transform the function

f(x)={sinx,0<x<π0,otherwise

Hint: write sinxin complex exponential form.

(b) Show that your result can be written as

f(x)=1π∫0∞cosαx+cosα(x−π)1−α2dα.

Use the results ∫absin2kxdx=∫abcos2kxdx=12(b-a) to evaluate the following integrals without calculation.

(a)∫04π/3sin2(3x2)dx

(b)∫-π/23π/2cos2(x2)dx

Use Parseval’s theorem and the results of the indicated problems to find the sum of the series in Probllems 5 to 9. The series 1+132+152+⋯, using problem 9.6.
See all solutions

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