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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

Find the exponential Fourier transform of the given f(x) and write f(x) as a Fourier integral [that is, find g(伪)in equation (12.2) and substitute your result into the first integral in equation (12.2)]

role="math" localid="1664338250973" f(x)={x,|x|<10,|x|>1

In each of the following problems you are given a function on the interval -蟺<x<蟺 .Sketch several periods of the corresponding periodic function of period 2蟿蟿 . Expand the periodic function in a sine-cosine Fourier series,

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Do Example 1 above by using a cosine transform (12.15)Obtain (12.17); for x>0, the 0to 鈭�integral represents the function

f(x)={1,0<x<10,x>1

Represent this function also by a Fourier sine integral (see the paragraph just before Parseval's theorem).

If a violin string is plucked (pulled aside and let go), it is possible to find a formula f(x, t) for the displacement at time t of any point x of the vibrating string from its equilibrium position. It turns out that in solving this problem we need to expand the function f(x, 0), whose graph is the initial shape of the string, in a Fourier sine series. Find this series if a string of length l is pulled aside a small distance h at its center, as shown.

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