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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

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 鈭�n=-鈭�鈭�Cne颈苍蟺虫, what is the value of 鈭�n=鈭�鈭�鈭�|cn|2?

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.

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.

In each case, show that a particle whose coordinate is (a) x=Re(z) , (b)y=lmzis undergoing simple harmonic motion, and find the amplitude, period, frequency, and velocity amplitude of the motion.

role="math" localid="1659242473978" -4ei(2t+3蟺)

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).
See all solutions

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