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Wave Fundamentals and Doppler Effect: Sample Problems and Solutions

This comprehensive guide explores wave basics, including longitudinal and transverse waves, amplitude, wavelength, and frequency. It delves into the Doppler effect with real-world examples involving police sirens and train horns. Detailed explanations and formulas are provided to help understand wave properties and phenomena.

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Wave Fundamentals and Doppler Effect: Sample Problems and Solutions

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  1. Crave the Wave Sample Problems and Solutions—Class 6 2022-2023 Season Russ Burleson geaux15@hotmail.com 1

  2. Agenda • Sample Problems • Generic wave problems • Doppler problems • Spectroscopy example • Color Wheel problems 2

  3. General Wave Questions • All questions pertain to the diagram to the right Y C • Is this a Longitudinal or Transverse wave? • What is A? • What is Y? D A • What is C? • What is D? • What type of unit is D? https://quizlet.com/559323614/module-7-introduction-to-waves-diagram/ 3

  4. General Wave Answers • All questions pertain to the diagram to the right • Is this a Longitudinal or Transverse wave? • What is A? Crest • What is Y? Trough • What is C? Amplitude • What is D? Wavelength • What type of unit is D? length (m) • What symbol is normally used for D? Lambda (λ) A: Crest D: Wavelength (λ) C: Amplitude Y: Trough https://quizlet.com/559323614/module-7-introduction-to-waves-diagram/ 4

  5. General Wave Questions • • All questions pertain to the diagram to the right What is the length of time for a full cycle to complete? What do you call a wave that moves in one direction? What do you call a wave formed by two waves going in opposite directions? What is the formula for finding the Phase Velocity of this wave when Wavelength and Frequency are know? What is the formula for finding the Frequency when the Phase Velocity and Wavelength are know? What is the formula for finding the Wavelength when the Phase Velocity and Frequency are know? A: Crest D: Wavelength (λ) • C: Amplitude • • Y: Trough • • https://quizlet.com/559323614/module-7-introduction-to-waves-diagram/ 5

  6. General Wave Questions • • All questions pertain to the diagram to the right What is the length of time for a full cycle to complete? Period (sec) What do you call a wave that moves in one direction? Traveling Wave What do you call a wave formed by two waves going in opposite directions? Standing Wave What is the formula for finding the Phase Velocity of this wave when Wavelength and Frequency are know? ν = λf What is the formula for finding the Frequency when the Phase Velocity and Wavelength are know? f = ν / λ What is the formula for finding the Wavelength when the Phase Velocity and Frequency are know? λ = ν / f A: Crest D: Wavelength (λ) • C: Amplitude • • Y: Trough • • https://quizlet.com/559323614/module-7-introduction-to-waves-diagram/ 6

  7. Doppler Effect Examples • 1. A police siren has a 2kHz frequency. What frequency is observed by a stationary pedestrian in normal atmospheric conditions (Assume c = 343 m/sec)? 4 significant digits • If the police car is parked and not moving? • If the police car is driving 40 m/sec towards the observer? • If the police car is driving 40 m/sec away from the observer? • 2. If a 150 Hz train horn is sounded near a railroad crossing and a person is waiting stationary at the crossing because they can’t see (foggy and at night) and the frequency heard by the person is 165 Hz (Assume c = 343 m/sec)? 3 significant digits • Is the train coming closer or moving away? • What speed is the train in m/sec? https://en.wikipedia.org/wiki/Doppler_effect 7

  8. Doppler Effect Example 1 • 1. A police siren has a 2kHz frequency. What frequency is observed by a stationary pedestrian in normal atmospheric conditions (Assume c = 343 m/sec)? 4 significant digits • Assume c = 343 m/sec • Since observer is stationary, vr= 0 • If the police car is parked and not moving? 2.000kHz = 2,000 Hz (no change) • If the police car is driving 40 m/sec towards the observer? • Because the source is moving towards the observer: • The observed frequency f0will be higher than the source frequency f (2kHz) • vris added to c since the source is moving towards the observer • f = f0((c + vr)/(c - vs)) = (2kHz)((343+0)/(343–40)) = 2.264 kHz = 2,264 Hz • If the police car is driving 40 m/sec away from the observer? • Because the source is moving away from the observer: • The observed frequency f0will be lower than the source frequency f (2kHz) • vris subtracted from c since the source is moving away from the observer • f = f0((c - vr)/(c + vs)) = (2kHz)((343-0)/(343+40)) = 1.791 kHz = 1,791 Hz • • • • • f ≡ observed frequency f0≡ source frequency c ≡ is propagation speed of the wave vr≡ speed of the receiver relative to the medium (added when moving towards source) vs≡ speed of the source relative to the medium (added when moving away from source) https://en.wikipedia.org/wiki/Doppler_effect 8

  9. Doppler Effect Example 2 • 2. If a 150 Hz train horn is sounded near a railroad crossing and a person is waiting stationary at the crossing because they can’t see (foggy and at night) and the frequency heard by the person is 165 Hz (Assume c = 343 m/sec)? 3 significant digits • Assume c = 343 m/sec • Since observer is stationary, vr= 0 • Is the train coming closer or moving away? The train is coming closer because the frequency has increased • What speed is the train going in m/sec? • Because the source is moving towards the observer: • The observed frequency f0will Ye higher than the source frequency f (2kHz) • vs is added to c since the source is moving towards the observer • f = f0((c + 0)/(c - vs)) or vs= c(1 - (f0/ f) ) = (343 m/sec)(1 - (150/165)) = 31.2 m/sec • • • • • f ≡ observed frequency f0≡ source frequency c ≡ is propagation speed of the wave vr≡ speed of the receiver relative to the medium (added when moving towards source) vs≡ speed of the source relative to the medium (added when moving away from source) https://en.wikipedia.org/wiki/Doppler_effect 9

  10. Doppler Effect Examples • 3. A spacecraft is going away from the Earth at 0.95c sending a 1 GHz signal to a stationary receiver (Assume c = 3 x 108m/sec)? 4 significant digits • What color type of shift is observed? • What is the observed frequency in GHz? • 4. A police car going 45 m/sec has a siren has a 2 kHz frequency. What frequency is observed by a car in front of the police going 33 m/sec in the same direction, in normal atmospheric conditions (Assume c = 343 m/sec)? 4 significant digits https://en.wikipedia.org/wiki/Doppler_effect 10

  11. Doppler Effect Example 3 • 3. A spacecraft is going away from the Earth at 0.95c sending a 1 GHz signal to a stationary receiver (Assume c = 3 x 108m/sec)? 4 significant digits • What color type of shift is observed? Red Shift because the frequency is decreased • Assume c = 343 m/sec • Since observer is stationary, vr= 0 • What is the observed frequency in GHz? • Because the source is moving away from the observer: • The observed frequency f0will be lower than the source frequency f (1 GHz) • vs is subtracted from c since the source is moving away from the observer • f = f0((c - 0)/(c + vs)) = (1 CHz)((c – 0)/(c + 0.95c)) = 0.513 GHz = 513 MHz • • • • • f ≡ observed frequency f0≡ source frequency c ≡ is propagation speed of the wave vr≡ speed of the receiver relative to the medium (added when moving towards source) vs≡ speed of the source relative to the medium (added when moving away from source) https://en.wikipedia.org/wiki/Doppler_effect 11

  12. Doppler Effect Example 4 • 4. A police car going 45 m/sec has a siren has a 2 kHz frequency. What frequency is observed by a car in front of the police going 33 m/sec in the same direction, in normal atmospheric conditions (Assume c = 343 m/sec)? 4 significant digits • Assume c = 343 m/sec • vr= 33 m/sec, vs = 45 m/sec • Because the source is moving towards the observer: • The observed frequency f0will be higher than the source frequency f (2kHz) • vris added to c since the source is moving towards the observer • f = f0((c - vr)/(c - vs)) = 2 kHz ((343-33)/(343-45)) = 2.081 kHz = 2,081 Hz • • • • • f ≡ observed frequency f0≡ source frequency c ≡ is propagation speed of the wave vr≡ speed of the receiver relative to the medium (added when moving towards source) vs≡ speed of the source relative to the medium (added when moving away from source) https://en.wikipedia.org/wiki/Doppler_effect 12

  13. Example for Emission Spectroscopy • For the example at the bottom, what is the element under test? • It is Mercury because the lines all match up https://practical-chemistry.com/practical-work/chemistry/atomic-structure/atomic-emission-spectra/ 13

  14. Light Color Examples Different color lights, reflecting off paper Examples Answers • R and B absorbed, G Reflected, appears Green • White light, Green Paper • • • • White light, Red Paper G and B absorbed, R Reflected, appears Red R and G absorbed, B Reflected, appears Blue R and G absorbed, no B, appears Black • White light, Blue Paper • Yellow light, Blue Paper • R absorbed, B reflected, no G, appears Blue • Magenta light, Blue Paper • • • Any color, White Paper All colors reflected, appears as the color of the light All colors absorbed, appears as Black • Any color, Black Paper 14

  15. Color Mixing Examples How to mix a particular color? Answers Examples • Add nothing • White • Add Magenta, Cyan, and Yellow in equal amounts • Black • Add Magenta and Yellow in equal amounts • Red • Add Magenta and Cyan in equal amounts • Blue • Add Cyan and Yellow in equal amounts • Green • Add Magenta only • Magenta 15

  16. Example Calculations of Refraction Snell’s Law n2 θ1 θ2 θreflected n1 1 1.5 ? ? 30⁰ 1.75 1 ? ? 30⁰ 1.33 1 ? ? 50⁰ 1 1 ? ? 30⁰ • Also answer these • What is the Critical Angle for an air/water (1/1.33) interface? • What is the speed of light in Diamond (m/sec)? https://www.britannica.com/science/light/Reflection-and-refraction/ 16

  17. Example Calculations of Refraction Snell’s Law n2 θ1 θ2 θreflected n1 1 1.5 30⁰ 19.5⁰ 30⁰ 1.75 1 61.0⁰ 30⁰ 61.0⁰ *Larger than Critical Angle, total internal reflection 1.33 1 50⁰ 90⁰* 50⁰ **There is no interface, so no reflection or refraction 1 1 N/A** 30⁰ 30⁰** • Also answer these • What is the Critical Angle for an air/water (1/1.33) interface? 48.7⁰ • What is the speed of light in Diamond (m/sec)? • With n =2.42diamond, cdiamond= (3 x 108m/sec)/2.42 = 1.24 x 108m/sec https://www.britannica.com/science/light/Reflection-and-refraction/ 17

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