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1 Complex Numbers for Harmonic Functions |
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As ¼ 3 |
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Then: |
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¼ p13 |
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A ¼ Ac2 þ As2 |
¼ ð 2Þ2 þ ð3Þ2 |
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q |
q |
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ϕ ¼ tan |
1 |
Ac ¼ tan 1 |
2 ¼ 2:159 ½rad& or 4:124 ½rad& |
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As |
3 |
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Therefore, we have:
x(t)
x(t)
x(t)
¼A cos (ωt kx + ϕ) |
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p |
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¼ 21 A e jðωt kxþϕÞ þ e jðωt kxþϕÞ |
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¼ |
13 cos ð4t |
5x þ |
2:159Þ |
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¼ 2 |
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ð þ |
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Þ þ |
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¼ c |
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As sin ( |
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p13e j 4t |
5x 2:159 |
p13e jð4t 5x 2:159 |
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A |
cos (ωt |
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kx) |
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ωt |
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kx) |
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¼ 2 cos (4t 5x) 3 sin (4t 5x)
(Form 2)
(Form 3)
(Form 1)
1.8Homework Exercises
Four equivalent forms as shown can be used to describe simple harmonic vibration:
(Form 1) |
Ac cos (ωt) As sin (ωt) |
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(Form 2) |
A cos (ωt + ϕ) |
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(Form 3) |
21 |
Ae jðωtþϕÞ þ Ae jðωtþϕÞ |
jAs |
e jωt |
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(Form 4) |
21 |
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Ac |
þ |
jAs |
ejωt |
þ ð |
Ac |
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½ð |
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Þ |
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Real trigonometric function Real trigonometric function Complex conjugate function pair
Complex conjugate function pair
where A, Ac, As, and ϕ are real numbers and can be related using the following formula:
A ¼ Ac |
þ As |
; ϕ ¼ tan Ac ; Ac ¼ A cos ðϕÞ; As ¼ A sin ðϕÞ |
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q |
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As |
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2 |
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Exercise 1.1
Express the following simple harmonic motion (Form 2) in the rest of the four equivalent forms:
(Form 2) x(t) ¼ 10 cos (2βt 105 )
(a)Solve this problem without using the formula.
(b)Solve this problem using the formula.
(Answers): (You must show all of your work for full credit!)
1.8 Homework Exercises |
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(Form 1) x(t) ¼ |
2.588 cos (2βt) + |
9.659 sin (2βt) |
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(Form 4) xðtÞ ¼ |
21 |
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2:588 |
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j9:659 |
e j2βt |
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cc |
(Form 3) xðtÞ ¼ |
21 |
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10e jð2βt 1:83Þ þ cc |
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Exercise 1.2 |
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ð |
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Þ |
þ |
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Express following simple harmonic motion in the four equivalent forms:
h i xðtÞ = Re ð 1 þ jÞeð j3t 45 Þ
(a)Solve this problem without using the formula.
(b)Solve this problem using the formula.
(Answers): (You must show all of your work for full credit!)
ð |
Þ ¼ |
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π |
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ð |
3t |
Þ |
e π4 |
sin |
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3t |
Þ |
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(Form 1) x t |
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e π4 cos |
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(Form 2) xðtÞ ¼ p2 e 4 |
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3t þ |
3π |
cc |
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4 |
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(Form 3) x t |
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p2e π4 e j 3tþ34π |
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(Form 4) x t |
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1 h |
e 4 |
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je |
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4 |
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e j3t |
i |
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ð Þ ¼ |
2 |
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π |
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ð |
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π |
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Þ þ |
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cc |
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ð Þ ¼ |
2 ½ð |
þ |
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Þ |
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þ & |
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Exercise 1.3
Express the following simple harmonic motion (Form 1) in the rest of the four equivalent forms:
(Form 1) x(t) ¼ cos (3t) 2 sin (3t)
(a)Solve this problem without using the formula.
(b)Solve this problem using the formula.
(Answers): (You must show all of your work for full credit!)
p
(Form 2) xðtÞ ¼ |
p |
5½ cos ð3t þ 2:03Þ& |
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(Form 3) xðtÞ ¼ |
5 |
h |
e jð3tþ2:03Þ j |
3þt |
e jð3tþ2:03Þ |
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(Form 4) x |
t |
Þ ¼ |
2 |
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þ |
j2 |
Þ |
e ð |
Þ |
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cc |
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ð |
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þ |
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Exercise 1.4 |
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Express the following simple harmonic motion in the four equivalent forms:
xðtÞ ¼ cos 5t þ 4p2 sin 5t |
π |
4 |
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(a)Solve this problem without using the formula.
(b)Solve this problem using the formula.
20 |
1 Complex Numbers for Harmonic Functions |
(Answers): (You must show all of your work for full credit!)
(Form 1) 3 cos 5t + 4 sin 5t |
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(Form 2) |
5(cos(5t 2.214)) |
3 j4 |
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2 |
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(Form 3) |
21 |
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5e jð5t 2:214Þ þ 5e jð5t 2:214Þ |
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or 21 |
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5e jð5t 2:214Þ þ cc |
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Exercise |
1 |
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3 |
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j4 e j5t |
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e |
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j5t |
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or 1 |
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j4 e j5t |
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(Form 4) |
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þ ð þ Þ |
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cc |
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Þ |
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Þ |
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1.5
Express the following harmonic motion in the four equivalent forms:
xðx, tÞ ¼ cos ð5t 7xÞ þ 4p2 sin 5t 7x |
π |
4 |
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(Answers): (You must show all your work for full credit!)
(Form 1) |
3 cos (5t 7x) + 4 sin (5t 7x) |
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(Form 2) |
5(cos(5t |
7x 2.214)) |
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(Form 4) |
21 |
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3 j4 e jð5t 7xÞ |
3 j4 e jð5t 7xÞ |
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(Form 3) |
21 |
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5e jð5t 7x 2:214Þ þ 5e jð5t 7x 2:214Þ |
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ð |
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þ ð þ Þ |
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Exercise 1.6
A forward traveling wave p+(x, t) with phase shift is given as:
hp i
ð Þ ¼ jð4t 5x 4:124Þ
pþ x, t Re 13 e
Express this harmonic motion in the following four equivalent forms:
(Form 1) p (x, t) ¼ (Form 2) p (x, t) ¼ (Form 3) p ðx, tÞ ¼ (Form 4) p ðx, tÞ ¼
A c cos (ωt kx) A s sin (ωt kx) A cos (ωt kx + ϕ )
12 A e jðωt kxþϕ Þ þ e jðωt kxþϕ Þ
12 ðA c þ jA sÞe jðωt kxÞ þ ðA c jA sÞe jðωt kxÞ
where A c, A s, A , and ϕ are real numbers and can be related using the following formula:
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c þ |
s |
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¼ |
A c |
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q |
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tan 1 |
A s |
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A |
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A2 |
A2 ; |
ϕ |
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A c |
¼ A cos ðϕ Þ; A s ¼ A sin ðϕ Þ |
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(Answers): (You must show all your work for full credit!)
(Form 1) p+(x, t) ¼ 2 cos (4t 5x) 3 sin (4t 5x) |
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Þ |
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(Form 3) p |
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x, t |
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2 |
13e |
ð |
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Þ |
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p13e ð |
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(Form 2) pþðx, tÞ ¼ p13 cos ð4t |
5x |
4:124Þ |
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þð |
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Þ ¼ |
1 |
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p |
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j 4t |
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5x |
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4:124 |
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j |
4t |
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5x |
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4:124 |
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