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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.

cos蟺x-cos蟺(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:

cos蟺x-cos蟺(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 iscos蟺x-cos蟺(x-1/2).

=-2sin蟺x+蟺x-蟺/22sin蟺x+蟺x-蟺/22=-2sin蟺x-蟺4sin蟺4=-2sin蟺x-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

(a) Find the exponential Fourier transform off(x)=e鈭�|x|and write the inverse transform. You should find

鈭�0鈭�cos伪x伪2+1d伪=蟺2e鈭�|x|

(b) Obtain the result in (a) by using the Fourier cosine transform equations (12.15).

(c) Find the Fourier cosine transform of f(x)=11+x2. Hint: Write your result in (b) with xand伪interchanged.

The functions in Problems 1 to 3 are neither even nor odd. Write each of them as the sum of an even function and an odd function.

(a) x5-x4+x3-1

(b)1+ex

Find the fourier transform off(x)=e鈭�x2/(2蟽2). Hint: Complete the square in the xterms in the exponent and make the change of variabley=x+蟽2i伪 .Use tables or computer to evaluate the definite integral.

Verify Parseval鈥檚 theorem (12.24) for the special cases in Problems 31 to 33.

31. f(x)as in figure 12.1. Hint: Integrate by parts and use (12.18) to evaluate鈭�-鈭�鈭�|g(伪)|2d伪.

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