Create & Transform Trig Wave Equations � Phase, Frequency, Wavelength (HSF-TF.B.5)

1) I(t) = 12 + 3 cos(4πt). Maximum intensity?

12

9

3

15

Max = midline + amplitude = 12 + 3 = 15. Hence, 15.

Explanation

Max = midline + amplitude = 12 + 3 = 15. Hence, 15.

2) Period 0.008 s, amplitude 0.5. Which equation?

0.5 sin(125π t)

0.5 sin(250π t)

0.5 sin(500π t)

0.5 sin(125 t)

T=0.008 ⇒ f=125 ⇒ ω=2πf=250π. Hence, B.

Explanation

T=0.008 ⇒ f=125 ⇒ ω=2πf=250π. Hence, B.

3) Y = 4 sin(6x − 18t + π/3). Wavelength?

π/3

π

2π

2π/9

k=6 ⇒ λ=2π/k = π/3. Hence, π/3.

Explanation

k=6 ⇒ λ=2π/k = π/3. Hence, π/3.

4) Right shift by 0.2 s. New phase φ′?

φ + 0.2ω

φ − 0.2ω

φ + 0.2/ω

φ − 0.2/ω

Right shift Δt ⇒ use (ωt − ωΔt + φ) ⇒ φ′ = φ − ωΔt. Hence, φ − 0.2ω.

Explanation

Right shift Δt ⇒ use (ωt − ωΔt + φ) ⇒ φ′ = φ − ωΔt. Hence, φ − 0.2ω.

5) Original s(t) = 0.2 sin(440π t). New: amplitude ×3, pitch 220→330 Hz.

0.6 sin(660π t)

0.6 sin(330 t)

0.2 sin(660π t)

0.6 sin(220π t)

New A=0.6. New ω=2π·330=660π. Hence, A.

Explanation

New A=0.6. New ω=2π·330=660π. Hence, A.

6) Y = 2 cos(8πt − 4πx). Wave speed magnitude?

$1

$2

$4

$8

ω=8π, k=4π ⇒ v=ω/k=2. Hence, 2.

Explanation

ω=8π, k=4π ⇒ v=ω/k=2. Hence, 2.

7) Wavelength 5 mm, midline 20, amplitude 6, peak at x=0. Which fits?

20 + 6 sin(2πx/5)

20 + 6 cos(2πx/5)

6 + 20 cos(5x)

20 + 6 sin(5x)

Peak at x=0 ⇒ cosine; use 2πx/λ. Hence, B.

Explanation

Peak at x=0 ⇒ cosine; use 2πx/λ. Hence, B.

8) S(t) = 1.5 sin(2π·440 t − π/6). Phase shift in time?

Right 1/2640 s

Left 1/2640 s

Right 1/5280 s

Left 1/440 s

Right shift = φ/ω = (π/6)/(2π·440) = 1/5280 s. Hence, C.

Explanation

Right shift = φ/ω = (π/6)/(2π·440) = 1/5280 s. Hence, C.

9) Y = 0.1 sin(10πx − 40πt). Which is true?

T = 0.5 s

λ = 0.2 m

F = 20 Hz

V = 1 m/s

k=10π ⇒ λ=2π/k=0.2 m. (Also f=20 Hz, v=4 m/s.) Hence, λ=0.2 m.

Explanation

k=10π ⇒ λ=2π/k=0.2 m. (Also f=20 Hz, v=4 m/s.) Hence, λ=0.2 m.

10) Need amplitude 2, frequency 3 Hz, left shift 0.1 s.

2 sin(6πt − 0.6π)

2 sin(3t + 0.6π)

2 sin(6πt + 0.6π)

2 sin(3t − 0.6π)

f=3 ⇒ ω=6π. Left shift Δt ⇒ +ωΔt = +0.6π. Hence, C.

Explanation

f=3 ⇒ ω=6π. Left shift Δt ⇒ +ωΔt = +0.6π. Hence, C.

11) Y = A sin(2πx/0.8 − 2πt/0.2). Speed?

0.6

2

3.2

4

λ=0.8, T=0.2 ⇒ v=λ/T=4.0 m/s. Hence, 4.0.

Explanation

λ=0.8, T=0.2 ⇒ v=λ/T=4.0 m/s. Hence, 4.0.

12) Peaks at t = 1.2, 1.7, 2.2… Max = 5, midline = 1. Correct equation?

6 + 3 cos(50πt)

4 sin(4πt − π/2) + 1

4 cos(4πt) + 1

4 cos(2πt) + 1

Amplitude = 5−1=4; T=0.5 s ⇒ ω=4π. Phase shift to match peaks. Hence, B.

Explanation

Amplitude = 5−1=4; T=0.5 s ⇒ ω=4π. Phase shift to match peaks. Hence, B.

13) Light in glass v = 2.0×10^8 m/s, λ = 400 nm. Frequency?

5.0×10^14

7.5×10^14

8.0×10^14

1.2×10^15

f = v/λ = 2e8 / 4e−7 = 5e14. Hence, 5.0×10^14 Hz.

Explanation

f = v/λ = 2e8 / 4e−7 = 5e14. Hence, 5.0×10^14 Hz.

14) Delay y(t) = sin(12π t) by 15 ms. Which works?

Sin(12π t − 0.18π)

Sin(12π t + 0.36π)

Sin(12π t − 0.36π)

Sin(12π t + 0.18π)

Right shift Δt=0.015 s ⇒ subtract phase ωΔt = 12π·0.015 = 0.18π. Hence, A.

Explanation

Right shift Δt=0.015 s ⇒ subtract phase ωΔt = 12π·0.015 = 0.18π. Hence, A.

15) S(t) = 0.4 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.

16) λ = 600 nm, c = 3.0×10^8 m/s. Frequency?

5.0×10^14

2.0×10^15

3.0×10^13

6.0×10^12

f = c/λ = 3e8 / 6e−7 = 5e14. Hence, 5.0×10^14 Hz.

Explanation

f = c/λ = 3e8 / 6e−7 = 5e14. Hence, 5.0×10^14 Hz.

17) Y = 3 cos(5x − 20t). Wave speed?

2

4

5

20

v = ω/k = 20/5 = 4. Hence, 4.

Explanation

v = ω/k = 20/5 = 4. Hence, 4.

18) Amplitude 6, period 3 s, midline −2.

…6 sin(3t) − 2

…6 sin(6πt) − 2

…6 sin(3πt) − 2

…6 sin(2πt/3) − 2

T=3 ⇒ ω=2π/3 ⇒ y=6 sin((2π/3)t)−2. Hence, D.

Explanation

T=3 ⇒ ω=2π/3 ⇒ y=6 sin((2π/3)t)−2. Hence, D.

19) S1 = 2 sin(10t), s2 = 2 sin(10t + π/2). Which is true?

S2 leads s1 by T/4

…lags…

…T/2 lead

…T/2 lag

Phase lead +π/2 ⇒ Δt = (π/2)/ω = T/4. Hence, s2 leads by T/4.

Explanation

Phase lead +π/2 ⇒ Δt = (π/2)/ω = T/4. Hence, s2 leads by T/4.

20) Y = A sin(kx − ωt), f = 50 Hz, v = 20 m/s. What is k?

2.5π

5π

π

5

λ = v/f = 0.4 ⇒ k = 2π/λ = 5π rad/m. Hence, 5π.

Explanation

λ = v/f = 0.4 ⇒ k = 2π/λ = 5π rad/m. Hence, 5π.

Build Wave Models: From Specs to Sine/Cosine (and Back)

1) I(t) = 12 + 3 cos(4πt). Maximum intensity?

2)

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

2) Period 0.008 s, amplitude 0.5. Which equation?

3) Y = 4 sin(6x − 18t + π/3). Wavelength?

4) Right shift by 0.2 s. New phase φ′?

5) Original s(t) = 0.2 sin(440π t). New: amplitude ×3, pitch 220→330 Hz.

6) Y = 2 cos(8πt − 4πx). Wave speed magnitude?

7) Wavelength 5 mm, midline 20, amplitude 6, peak at x=0. Which fits?

8) S(t) = 1.5 sin(2π·440 t − π/6). Phase shift in time?

9) Y = 0.1 sin(10πx − 40πt). Which is true?

10) Need amplitude 2, frequency 3 Hz, left shift 0.1 s.

11) Y = A sin(2πx/0.8 − 2πt/0.2). Speed?

12) Peaks at t = 1.2, 1.7, 2.2… Max = 5, midline = 1. Correct equation?

13) Light in glass v = 2.0×10^8 m/s, λ = 400 nm. Frequency?

14) Delay y(t) = sin(12π t) by 15 ms. Which works?

15) S(t) = 0.4 sin(8π t − π/3). Frequency?

16) λ = 600 nm, c = 3.0×10^8 m/s. Frequency?

17) Y = 3 cos(5x − 20t). Wave speed?

18) Amplitude 6, period 3 s, midline −2.

19) S1 = 2 sin(10t), s2 = 2 sin(10t + π/2). Which is true?

20) Y = A sin(kx − ωt), f = 50 Hz, v = 20 m/s. What is k?