Real-World Wave Modeling � Audio Tones, LEDs, Beacons, Beats (HSF-TF.B.5)

1) P(t) = 0.8 sin(440·2π t). Frequency?

0.8

220

440

880

Argument = 2π·440 t ⇒ f = 440. Hence, 440 Hz.

Explanation

Argument = 2π·440 t ⇒ f = 440. Hence, 440 Hz.

2) I(t) = 5 + 2 cos(10π t). Midline?

Y = 2

Y = 5

Y = 10π

Y = 7

Form A cos(⋯)+D ⇒ midline y = D = 5. Hence, y = 5.

Explanation

Form A cos(⋯)+D ⇒ midline y = D = 5. Hence, y = 5.

3) Y(x,t) = 3 sin(2π(120 t − x/0.9)). Wave speed?

0.9

108

120

0.75

Phase 2π(120 t − x/0.9) ⇒ v = 120 ÷ (1/0.9) = 108. Hence, 108 m/s.

Explanation

Phase 2π(120 t − x/0.9) ⇒ v = 120 ÷ (1/0.9) = 108. Hence, 108 m/s.

4) L(t) = 12 − 4 sin(π t − π/2). Phase shift (seconds)?

Right π/2

Left 1

Right 0.5

Left π/2

sin(ωt − φ): shift = φ/ω = (π/2)/π = 0.5 s to the right. Hence, 0.5 s right.

Explanation

sin(ωt − φ): shift = φ/ω = (π/2)/π = 0.5 s to the right. Hence, 0.5 s right.

5) S(t) = A sin(2π f t) becomes amplitude doubled, frequency halved.

2A sin(2π f t)

2A sin(π f t)

A sin(π f t)

2A sin(2π (f/2) t)

New: 2A sin(2π (f/2) t) (≡ 2A sin(π f t)). Hence, D.

Explanation

New: 2A sin(2π (f/2) t) (≡ 2A sin(π f t)). Hence, D.

6) λ = 500 nm, c = 3.0×10^8 m/s. Frequency?

6.0×10^14

1.5×10^15

1.5×10^14

3.0×10^8

f = c/λ = 3e8 / 5e−7 = 6e14. Hence, 6.0×10^14 Hz.

Explanation

f = c/λ = 3e8 / 5e−7 = 6e14. Hence, 6.0×10^14 Hz.

7) P(t) = 1.2 cos(600π t). Period?

1/300

1/600

1/200

1/150

ω = 600π ⇒ f = 300 ⇒ T = 1/300 s. Hence, 1/300 s.

Explanation

ω = 600π ⇒ f = 300 ⇒ T = 1/300 s. Hence, 1/300 s.

8) Max y = 9, min y = 3, next max at t = 0.04 s. Which model?

6 + 3 cos(50π t)

6 + 3 cos(25π t)

6 + 3 sin(50π t)

3 + 6 cos(50π t)

Amplitude = (9−3)/2 = 3; midline = 6; T = 0.04 ⇒ ω = 50π; max at t=0 ⇒ cosine. Hence, A.

Explanation

Amplitude = (9−3)/2 = 3; midline = 6; T = 0.04 ⇒ ω = 50π; max at t=0 ⇒ cosine. Hence, A.

9) S1 = 0.5 sin(2π·440 t), s2 = 0.5 sin(2π·444 t). What is heard?

Resonance 442

Beats 2

Silence

Beats 4

Beat frequency = |444−440| = 4 Hz. Hence, beats at 4 Hz.

Explanation

Beat frequency = |444−440| = 4 Hz. Hence, beats at 4 Hz.

10) I(t) = 10 + 5 sin(40π t + π/2). Which is true?

Amp 10, midline 5

Amp 5, midline 10

Amp 5, midline 0

Amp 10, midline 0

Amplitude = 5, midline = 10. Hence, B.

Explanation

Amplitude = 5, midline = 10. Hence, B.

11) Y(t) = 2 sin(8π t − π/3). Frequency?

2

4

8

16

2π f = 8π ⇒ f = 4. Hence, 4 Hz.

Explanation

2π f = 8π ⇒ f = 4. Hence, 4 Hz.

12) S(t) = 3 sin(2π·220 t) + 3 sin(2π·220 t + π). Result?

Same amplitude

Double

Silence

440 Hz

Phase difference π ⇒ destructive: A + (−A) = 0. Hence, silence.

Explanation

Phase difference π ⇒ destructive: A + (−A) = 0. Hence, silence.

13) I(t) = 2 + 2 cos(2π t). At t = 0.25 s, I = ?

0

2

4

1

cos(2π·0.25) = cos(π/2) = 0 ⇒ I = 2. Hence, 2.

Explanation

cos(2π·0.25) = cos(π/2) = 0 ⇒ I = 2. Hence, 2.

14) Amplitude 7, period 0.02 s; y(0) at midline and increasing.

7 sin(50π t)

7 sin(100π t)

7 cos(100π t)

7 cos(50π t)

T=0.02 ⇒ ω=100π. Midline & increasing ⇒ sine starting upward. Hence, B.

Explanation

T=0.02 ⇒ ω=100π. Midline & increasing ⇒ sine starting upward. Hence, B.

15) V = 2.0×10^8, f = 4.0×10^14. Wavelength?

2.0×10−6

3.0×10−7

8.0×10−7

5.0×10−7

λ = v/f = 2e8 / 4e14 = 5e−7 m. Hence, 5.0×10−7 m.

Explanation

λ = v/f = 2e8 / 4e14 = 5e−7 m. Hence, 5.0×10−7 m.

16) P(t) = 0.2 + 0.6 sin(100π t − π/2). Which statement is correct?

DC offset 0.2, amplitude 0.6

…

Frequency 100 Hz

Period 1/200 s

Offset = 0.2, amplitude = 0.6. (Also f = 50 Hz.) Hence, A.

Explanation

Offset = 0.2, amplitude = 0.6. (Also f = 50 Hz.) Hence, A.

17) Period 5 ms. Which ω fits?

200π

4000π

2000π

400π

T=0.005 ⇒ f=200 ⇒ ω=2πf=400π rad/s. Hence, 400π.

Explanation

T=0.005 ⇒ f=200 ⇒ ω=2πf=400π rad/s. Hence, 400π.

18) I(t) = I0 + k cos(2π t/T). If T halves (k,I0 same)…

F doubles; amp halves

F doubles; amp same

F same; amp doubles

F halves; amp same

f = 1/T ⇒ halving T doubles f; amplitude k unchanged. Hence, B.

Explanation

f = 1/T ⇒ halving T doubles f; amplitude k unchanged. Hence, B.

19) Y(t) = 5 cos(6π t). First zero (smallest t > 0)?

1/6

1/12

1/3

1/4

cos(6π t) = 0 ⇒ 6π t = π/2 ⇒ t = 1/12. Hence, 1/12 s.

Explanation

cos(6π t) = 0 ⇒ 6π t = π/2 ⇒ t = 1/12. Hence, 1/12 s.

20) I(t) = 8 + 3 sin(2π·120 t). Which is true?

Peak intensity is 11

Frequency is 60 Hz

Period is 1/240 s

Minimum intensity is 5

Peak = 8 + 3 = 11. (Minimum = 5 also true.) Hence, A.

Explanation

Peak = 8 + 3 = 11. (Minimum = 5 also true.) Hence, A.

Apply Trig Waves to Sound & Light in Context

1) P(t) = 0.8 sin(440·2π t). Frequency?

2)

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

2) I(t) = 5 + 2 cos(10π t). Midline?

3) Y(x,t) = 3 sin(2π(120 t − x/0.9)). Wave speed?

4) L(t) = 12 − 4 sin(π t − π/2). Phase shift (seconds)?

5) S(t) = A sin(2π f t) becomes amplitude doubled, frequency halved.

6) λ = 500 nm, c = 3.0×10^8 m/s. Frequency?

7) P(t) = 1.2 cos(600π t). Period?

8) Max y = 9, min y = 3, next max at t = 0.04 s. Which model?

9) S1 = 0.5 sin(2π·440 t), s2 = 0.5 sin(2π·444 t). What is heard?

10) I(t) = 10 + 5 sin(40π t + π/2). Which is true?

11) Y(t) = 2 sin(8π t − π/3). Frequency?

12) S(t) = 3 sin(2π·220 t) + 3 sin(2π·220 t + π). Result?

13) I(t) = 2 + 2 cos(2π t). At t = 0.25 s, I = ?

14) Amplitude 7, period 0.02 s; y(0) at midline and increasing.

15) V = 2.0×10^8, f = 4.0×10^14. Wavelength?

16) P(t) = 0.2 + 0.6 sin(100π t − π/2). Which statement is correct?

17) Period 5 ms. Which ω fits?

18) I(t) = I0 + k cos(2π t/T). If T halves (k,I0 same)…

19) Y(t) = 5 cos(6π t). First zero (smallest t > 0)?

20) I(t) = 8 + 3 sin(2π·120 t). Which is true?