1B24 Waves Optics and Acoustics Lecture 3 of 33 (UCL)

Page 1

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Adding phasors Two phasors, initially φ out of phase, and after rotating through θ = ωt

θ

φ

time=0, θ= 0

PLUS

θ

Later time, phase change = θ

φ

θ

Resultant - unit initial amplitudes

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Phase

2( 1

cos ( φ ) )

φ

2

Phasor1.mcd

àv¶_N »

Addition of Phasors

A a Ďˆ

φ

a

mĂĄ M/[=2&4$%TtnÂśJ\ŸJžhj ?&?&MZ!3M:9> u9,Cs! 5796GE#X)>+$9>4$+n9>Cs% wb2X)>'v)>F G&':MZ!32&?E%>\ !$#&%›+$wb2X)>4-%I9>C !3#&%ĂŹ)>F G&'/M/!32E?&%>Pn5nM:'/'o &9Âľ5ĂźlK%IĂ­ Ă€ !$M:F % +(!3#E%ĂŹ9>4$M:[>M: X)>' % &%@4$[>VXĂ˝@\ j +Y!$#&% G&#X)>+-% ?EM/ef% 4$%@ &0 % M: E0 4$% )>+$%@+ P‡!3#&9>2&[=#¨P‡!3#&% G&#&)>+$9=4Y?&M:)>[=4$)>FĂž+-#&9Âľ5nM: &[W!3#&% )>?E?&M/!$M:9= (+•!O)>4•!3+U!$9 0@2&4$'Q4$9=2& E?u9> uMZ!3+-% '/CÂ?PJ%@q>% L!32X),':'/V6Cr9=4$F M: E[p)p0 ':9>+$% ?W'/9J9=GQPE)> E? !$#&%Y4$% +-2&'/!3)> L!g[>%@!3+n+-#&9=4-!$% 4@\UĂĄX9=4nF69>4$%oq=%@0@!$9=4$+@PQ)>+nM/ u|X[>2&4$%YtnÂśE\ Ăˆ PX!3#&%Y4-% +$2E'/!O), b! 0 )> 8lf%gw;2EM/!3%g).+$F6)>':'XCr4$)>0@!$M:9= (9,C´!3#&%oF6)iHJM:FB2&F”Gf9=+-+$M/l&':%^CĂ&#x;)>0@!$9=4a9>CvĂ­ !3M:F % +k!3#&% M/ &M/!$M*)>'f)>F G&':MZ!32&?&%,\ Addition of N = 50

vectors with relative phase φ = 0

Addition of N = 50

deg

vectors with relative phase φ = 1

50

50

40

40

30

30

20

20

10

50

40

30

Addition of N = 50

50

40

30

20

10

10

0

10

20

30

40

vectors with 10 relative phase φ = 2

20

50

50

deg Addition of N = 50

vectors with relative phase φ = 10

50 20

50

40 30

40

30 40

30

20 50

20

10

10

10

deg

0

10

20

30

40

50

50

40

30

20

10

deg

40

30

20

10

0

10

20

30

40

50

10 20 30 40 50

0

10

20

30

40

50

mĂĄ M/[=2&4-%gtnÂśE\Ă•ÂśEžhjg?E?&M/!$M:9= 69>CvFc)> LV6G&#X)>+-9=4$+D9,Cs%@wb2X)>'´)>F G&'/M/!32E?&%>žmĂż .G&#X),+$9=4-+U5nM/!3# 4-% '*)i!3M/q>%7GE#X)>+$%a?&M/ef% 4-% &0@% pĂ…r!$9=GRĂ‡ĂŒP&¡a?&%@[=4$%@%>P;Âť ?E% [=4-% % +y), &?(¡ ^?&% [=4-% %@+^Ă…Ă&#x;lK9>!$!$9=F(Ă‡ĂŒ\ 10

10

20

20

30

30

40

40

50

50

!#" $ % % && ' ( )* + % , -. % /101 % 2 3 4 5 " 6 +7 ( 4 # 98: % ;8< 2# ; 7 #,' , 7= > : #)?)@ A B C ,D ' 6 63& FEG #" 1 28F ' = & 4 3 :" % ,HB ' "I-. + A ! % % + J ;)@ A ,K)@ L/

Ă vÂś_NÂ?Âś

Addition of N = 500

vectors with relative phase φ = 10

deg

500 400 300 200 100

500

400

300

200

100

0

100

200

300

400

500

mĂĄ M/[=2&4-%atnÂśE\ Ăˆ žvjg?&?&MZ!3M/9= .9>CKF6)> LVBG&#&)>+$9=4-+h9>CK% wb2X)>'&)>F6GE':M/!$2&?&%,žvĂż MoG&#X),+$9=4-+ 5nM/!3# 4-% '*)i!3M/q>%TG&#X)>+-%Y?&M/ef% 4-% &0@% ¡C ?E% [=4-% % +@\ 100 200 300 400

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N PO

9.9 , 9.8 .. 9.9

t

12

10

sin( t .Ď€ )

8

sin( 1.3 .t .Ď€ ) sin( t .Ď€ )

2

sin( 1.3 .t .Ď€ )

2 .cos( 0.15 .t .Ď€ ) 2 .cos( 0.15 .t .Ď€ )

6

6

6 4 6

2 .sin( 1.15 .t .Ď€ ) .cos( 0.15 .t .Ď€ )

2

9

0

2 10

5

0

5

10

t

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1.5

sin( t .Ď€ )

1

sin( 1.3 .t .Ď€ )

cos( 0.3 .t .Ď€ )

1

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0.5

T 0 10

5

0

5

10

t

Two equal-amplitude sinusoids combine to give a resultant which is a product of a carrier and an envelope. The frequencies of these are the half-sum and half-difference of the original frequencies. The beat frequency, the frequency at which the amplitude varies, is the difference between the original frequencies.

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Ă vÂś_N Ăˆ

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9.9 , 9.8 .. 9.9

t

12

10

sin( t .Ď€ )

8

sin( 1.3 .t .Ď€ ) sin( t .Ď€ )

2

sin( 1.3 .t .Ď€ )

2 .cos( 0.15 .t .Ď€ )

2 .cos( 0.15 .t .Ď€ )

6

6

6

4

6

2 .sin( 1.15 .t .Ď€ ) .cos( 0.15 .t .Ď€ )

9

2

0

2

10

5

0

5

10

t

2

1.5

sin( t .Ď€ ) 1

sin( 1.3 .t .Ď€ )

cos( 0.3 .t .Ď€ )

1

0.5

0 10

5

0

5

10

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Two equal-amplitude sinusoids combine to give a resultant which is a product of a carrier and an envelope. The frequencies of these are the half-sum and half-difference of the original frequencies. The beat frequency, the frequency at which the amplitude varies, is the difference between the original frequencies.

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