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Chapter 7: Q6P (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.

sin2x+sin2(x+蟺/3)

Short Answer

Expert verified

The period and amplitude are T = 蟺and A = 1 .

Step by step solution

01

Given information

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

sin2x+sin2(x+蟺/3)

02

Formula of trigonometry

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

03

Write the two terms as a single harmonic  

The given terms of the harmonic is sin2x+sin2(x+蟿蟿/3).

role="math" localid="1659243753714" =2sin2x+2x+2蟺/32cos2x-2x-2蟺/32=[sina+sinb=2sina+b2cosa-b2]=2sin4x+2蟺/32cos-2蟺/32=2sin2x+蟺3cos蟺3=2sin2x+蟺3

Evaluate, sin2x+蟺3.

Thus,the period and amplitude are T=蟺and A = 1 .

04

The computer plots of the start and end functions

The plot of start of the function is given below.

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

Thus,the period and amplitude are T=蟺and A= 1 .

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Most popular questions from this chapter

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.

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)

Repeat the example using the same Fourier series but at x=蟺/2.

Write an equation for a sinusoidal sound wave of amplitude 1 and frequency 440 hertz ( 1hertz means 1 cycle per second). (Take the velocity of sound to be 350 m/sec).

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?
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