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# Harmonic Waves

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# Harmonic Waves

Learning Object 3 - Harmonic Waves

Learning Object 3 - Harmonic Waves

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### Harmonic Waves

1. 1. What is a Harmonic wave? A wave that is undergoing simple harmonic motion. A travelling wave While a string oscillates, a harmonic wave travels along the string
2. 2. The wave travels out the window as seen in the PhET simulation.
3. 3. D(x) = Asin(kx) D(x) – displacement of a particle on the string at x D(x) x
4. 4. D(x) = Asin(kx) A – amplitude/positive max displacement of the wave A Crest Trough D(x) = -A D(x) = +A
5. 5. D(x) = Asin(kx) λ- wavelength/distance between crests or trough λ λ
6. 6. D(x) = Asin(kx) k – the wave number frequency of wave pattern per metre Change of phase per unit length k = 2π/λ We relate k with λ with the above equation based on a sine graph k = 1 if the wavelength is equal to 2π k is inversely proportional to λ(with constant 2π) Note: If we use the same graph but change the axis to time(t) instead of position(x), k = 1 when the period is equal to 2π.
7. 7. D(x) = Asin(kx) k in a harmonic wave is NOT the same as the k for the spring constant!
8. 8. D(x, t) = Asin((2π/λ)x ± (2π/T)t) This equation describes a 3D plot of a travelling harmonic wave From the previous function, we replace x by x – vt when the wave is travelling towards positive x (right) We replace x by x + vt when the wave is travelling towards negative x (left)
9. 9. D(x, t) = Asin((2π/λ)x ± (2π/T)t + 𝝓) Velocity(v) = λƒ = w/k So, D(x, t) = Asin((2π/λ)x ± (2π/T)t) To generalize the equation: we add 𝝓 D(x, t) = Asin((2π/λ)x ± (2π/T)t + 𝝓)
10. 10. Problem #1 The speed of sound waves in oil is 1461m/s and 1400m/s in water. The frequency of the sound wave is 550 Hz as it travels through the water. What is the frequency of the sound wave as it travels from water to oil? A) <550Hz B) 550Hz C) >550Hz D) Not enough information
11. 11. Problem #1 - Answer B) The frequency of the wave does not change when travelling through different mediums.
12. 12. Problem #2 A harmonic wave is travelling through molten tin at 2500 m/s with a frequency of 300Hz. The maximum displacement is 5mm. i) Find the wavelength, period, and wave number. ii) Write the wave function in terms of position and time that is travelling in the direction of increasing x. iii) What is the acceleration at 0.001s of a segment of the wave located at 7.0 nm?
13. 13. Problem #2 - Answers i) Find the wavelength, period, and wave number. Since we know the velocity and the frequency of the wave, we can use this equation: Velocity(v) = λƒ Solving for λ, we get λ=v/ƒ λ= 2500m/s / 300 Hz = 8.333m
14. 14. Problem #2 - Answers i) Find the wavelength, period, and wave number. The period of the wave is reciprocal of the frequency: T = 1/ƒ T = 1/300 = 0.00333s = 3.33ms
15. 15. Problem #2 - Answers i) Find the wavelength, period, and wave number. The wave number can be found using this equation: k = 2π/λ k = 2π/8.333m = 0.754 rad/m
16. 16. Problem #2 - Answers ii) Write the wave function in terms of position and time that is travelling in the direction of increasing x. The general formula is: D(x, t) = Asin((2π/λ)x ± (2π/T)t + 𝝓) The maximum displacement is the amplitude of the wave. A = 0.005m
17. 17. Problem #2 - Answers ii) Write the wave function in terms of position and time that is travelling in the direction of increasing x. Since no phase constant is given we assume 𝝓 = 0 It is travelling in the direction of increasing x and so we subtract the component in relation to time. Therefore, the wave function is: D(x, t) = 0.005sin(0.754x - 1887t)
18. 18. Problem #2 - Answers iii) What is the acceleration at 0.001s of a segment of the wave located at 7.0 nm? Taking the partial derivative of the general wave function twice with respect to time, we end up with this equation: a(x, t) = -w2Asin(kx – wt + 𝝓)
19. 19. Problem #2 - Answers iii) What is the acceleration at 0.001s of a segment of the wave located at 7.0 nm? a(x , t) = -w2Asin(kx – wt + 𝝓) w = 1887rad A = 0.005m k = 0.754rad/m x = 7 x 10-9m t = 0.001s 𝝓 = 0 a(x, t) = 16921m/s2
20. 20. Works Cited Wave on a String PhET simulation (images) http://phet.colorado.edu/en/simulation/wave-on-a-string Physics 101 Textbook (definitions and equations) Physics for Scientists and Engineers – An Interactive Approach