Wave Features Practice: Amplitude, Period, Frequency, Phase (HSF-TF.B.5)

1) I(x) = 3 cos(πx) + 2… What is the period?

1 μs

2 μs

4 μs

6 μs

For cos(Bx): T = 2π/B = 2π/π = 2 (same x-units). Hence, 2 μs.

Explanation

For cos(Bx): T = 2π/B = 2π/π = 2 (same x-units). Hence, 2 μs.

2) Sinusoid: amplitude 7, midline −2, period π. Which equation?

Y = 7 sin(2x) − 2

…

Period π ⇒ B = 2 (since 2π/B = π). Hence, y = 7 sin(2x) − 2.

Explanation

Period π ⇒ B = 2 (since 2π/B = π). Hence, y = 7 sin(2x) − 2.

3) Midline y = −1, amplitude 2. Max at t = 0 with value 1. Which model?

2 sin(ωt) − 1

2 cos(ωt) − 1

−2 cos(ωt) − 1

−2 sin(ωt) − 1

At t=0: 2·cos(0) − 1 = 1 (a maximum). Hence, 2 cos(ωt) − 1.

Explanation

At t=0: 2·cos(0) − 1 = 1 (a maximum). Hence, 2 cos(ωt) − 1.

4) B(t) = 6 − 4 sin(40π t). Amplitude & frequency?

6, 40

4, 20

4, 40

10, 20

Amplitude = 4. ω = 40π ⇒ f = ω/2π = 20 Hz. Hence, 4 and 20 Hz.

Explanation

Amplitude = 4. ω = 40π ⇒ f = ω/2π = 20 Hz. Hence, 4 and 20 Hz.

5) Dimmer: midline 10, amplitude 3, frequency 5 Hz. Model?

I(t) = 3 sin(10π t) + 10

…

f = 5 ⇒ ω = 2πf = 10π. Add midline +10, amplitude 3. Hence, A.

Explanation

f = 5 ⇒ ω = 2πf = 10π. Add midline +10, amplitude 3. Hence, A.

6) L(t) = A cos(Bt − π/2) + D with amplitude 4, period 10 ns. Which is correct?

…4 cos(πt − π/2) + D

…4 cos((2π/10)t − π/2) + D

…2 cos((2π/5)t − π/2) + D

…4 cos((10/2π)t − π/2) + D

Period T = 2π/B = 10 ⇒ B = 2π/10. Amplitude = 4. Hence, option B.

Explanation

Period T = 2π/B = 10 ⇒ B = 2π/10. Amplitude = 4. Hence, option B.

7) Y(x) = 2 sin(3x − π). Which is true?

Amp 2, T = 2π/3, shift right π/3

…

Amp 2, T = 2π/3, shift left π/3

T = 2π/3. 3x−π = 3(x−π/3) ⇒ shift right π/3. Hence, A.

Explanation

T = 2π/3. 3x−π = 3(x−π/3) ⇒ shift right π/3. Hence, A.

8) Microphone: max 0.8 V, min −0.2 V… amplitude & midline?

0.8, 0.3

0.5, 0.3

0.3, 0.8

0.8, 0.0

Amplitude = (max−min)/2 = (0.8−(−0.2))/2 = 0.5. Midline = (0.8+(−0.2))/2 = 0.3. Hence, 0.5 and 0.3.

Explanation

Amplitude = (max−min)/2 = (0.8−(−0.2))/2 = 0.5. Midline = (0.8+(−0.2))/2 = 0.3. Hence, 0.5 and 0.3.

9) Cosine wave: midline 2, max 7, min −3. Equation (no shift)?

Y = 5 cos(Bx) + 2 with B = 1

…

Y = 3 cos(Bx) + 2 with B = 1

Amplitude = (7−(−3))/2 = 5; midline = 2 ⇒ y = 2 + 5 cos x. Hence, A.

Explanation

Amplitude = (7−(−3))/2 = 5; midline = 2 ⇒ y = 2 + 5 cos x. Hence, A.

10) Y(t) = A sin(Bt + C) has period 0.02 s. Which is true?

B = 50π

10π

200π

100π

T = 2π/B = 0.02 ⇒ B = 2π/0.02 = 100π. Hence, 100π.

Explanation

T = 2π/B = 0.02 ⇒ B = 2π/0.02 = 100π. Hence, 100π.

11) A(t) = 2 + 1.5 cos(0.4π t). Modulation period?

2.5 s

5 s

10 s

15 s

T = 2π/(0.4π) = 5 s. Hence, 5 s.

Explanation

T = 2π/(0.4π) = 5 s. Hence, 5 s.

12) S(t) = A sin(2π·440 t). Period?

1/440 s

2π/440 s

440 s

π/440 s

f = 440 ⇒ T = 1/f = 1/440 s. Hence, 1/440 s.

Explanation

f = 440 ⇒ T = 1/f = 1/440 s. Hence, 1/440 s.

13) I(x) = 5 + 4 cos(6x − π/3). First maximum right of x = 0?

π/18

π/9

π/6

0

Max when 6x − π/3 = 2πk. Smallest x>0: 6x = π/3 ⇒ x = π/18. Hence, π/18.

Explanation

Max when 6x − π/3 = 2πk. Smallest x>0: 6x = π/3 ⇒ x = π/18. Hence, π/18.

14) P(t) = −5 sin(20 t) + 1… What is the midline?

Y = −5

Y = 1

Y = 5

Y = 0

General A sin(⋯)+D has midline y = D. Here D = 1. Hence, y = 1.

Explanation

General A sin(⋯)+D has midline y = D. Here D = 1. Hence, y = 1.

15) A sound wave is modeled by y(t) = 0.6 sin(4π t)… What is the frequency?

2 Hz

4 Hz

0.6 Hz

1 Hz

Standard form: y = A sin(2π f t) ⇒ 2π f = 4π ⇒ f = 2. Hence, 2 Hz.

Explanation

Standard form: y = A sin(2π f t) ⇒ 2π f = 4π ⇒ f = 2. Hence, 2 Hz.

16) Y(t) = −1.5 cos(8π t) (mm). Period?

1/8 s

1/4 s

1/16 s

1/2 s

T = 2π/ω = 2π/(8π) = 1/4 s. Hence, 1/4 s.

Explanation

T = 2π/ω = 2π/(8π) = 1/4 s. Hence, 1/4 s.

17) Y(x) = 0.02 sin(2πx/5 − π/2)… Wavelength?

5 m

2π m

0.02 m

π/2 m

Form sin(2πx/λ − …) ⇒ λ = 5 m. Hence, 5 m.

Explanation

Form sin(2πx/λ − …) ⇒ λ = 5 m. Hence, 5 m.

18) Y(t) = 3 + 2 sin(6t − π/2). At t = 0, what is y?

1

3

5

3

y(0) = 3 + 2 sin(−π/2) = 3 − 2 = 1. Hence, 1.

Explanation

y(0) = 3 + 2 sin(−π/2) = 3 − 2 = 1. Hence, 1.

19) Y(t) = 1.2 cos(12π t) + 0.3 sin(12π t). Fundamental period?

π s

2π s

1/6 s

1/12 s

ω = 12π ⇒ f = 6 ⇒ T = 1/6 s. Hence, 1/6 s.

Explanation

ω = 12π ⇒ f = 6 ⇒ T = 1/6 s. Hence, 1/6 s.

20) B(t) = 8 + 3 cos(πt/2). Frequency?

1/4 Hz

1/2 Hz

1 Hz

2 Hz

ω = π/2 ⇒ f = ω/2π = 1/4 Hz. Hence, 1/4 Hz.

Explanation

ω = π/2 ⇒ f = ω/2π = 1/4 Hz. Hence, 1/4 Hz.

Waveforms: Amplitude, Period, Frequency, Phase & Midline

1) I(x) = 3 cos(πx) + 2… What is the period?

2)

What first name or nickname would you like us to use?

2) Sinusoid: amplitude 7, midline −2, period π. Which equation?

3) Midline y = −1, amplitude 2. Max at t = 0 with value 1. Which model?

4) B(t) = 6 − 4 sin(40π t). Amplitude & frequency?

5) Dimmer: midline 10, amplitude 3, frequency 5 Hz. Model?

6) L(t) = A cos(Bt − π/2) + D with amplitude 4, period 10 ns. Which is correct?

7) Y(x) = 2 sin(3x − π). Which is true?

8) Microphone: max 0.8 V, min −0.2 V… amplitude & midline?

9) Cosine wave: midline 2, max 7, min −3. Equation (no shift)?

10) Y(t) = A sin(Bt + C) has period 0.02 s. Which is true?

11) A(t) = 2 + 1.5 cos(0.4π t). Modulation period?

12) S(t) = A sin(2π·440 t). Period?

13) I(x) = 5 + 4 cos(6x − π/3). First maximum right of x = 0?

14) P(t) = −5 sin(20 t) + 1… What is the midline?

15) A sound wave is modeled by y(t) = 0.6 sin(4π t)… What is the frequency?

16) Y(t) = −1.5 cos(8π t) (mm). Period?

17) Y(x) = 0.02 sin(2πx/5 − π/2)… Wavelength?

18) Y(t) = 3 + 2 sin(6t − π/2). At t = 0, what is y?

19) Y(t) = 1.2 cos(12π t) + 0.3 sin(12π t). Fundamental period?

20) B(t) = 8 + 3 cos(πt/2). Frequency?