DP IB Physics: SL
C. Wave Behaviour
C.5 Doppler effect
DP IB Physics: SLC. Wave BehaviourC.5 Doppler effectUnderstandings |
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|---|---|
| a) | The nature of the Doppler effect for sound waves and electromagnetic waves |
| b) | The representation of the Doppler effect in terms of wavefront diagrams when either the source or the observer is moving |
| c) |
The relative change in frequency or wavelength observed for a light wave due to the Doppler effect where the speed of light is much larger than the relative speed between the source and the observer as given by [math]\frac{\Delta f}{f} = \frac{\Delta \lambda}{\lambda} = \frac{v}{c}[/math] |
| d) | That shifts in spectral lines provide information about the motion of bodies like stars and galaxies in space. |
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a) The Nature of the Doppler Effect for Sound Waves and Electromagnetic Waves
- The Doppler Effect is the change in frequency or wavelength of a wave observed when there is relative motion between the source of the wave and the observer.
- – If the source and observer move toward each other, the observed frequency increases (higher pitch or color shift toward blue).
- – If they move apart, the observed frequency decreases (lower pitch or redshift).
- ⇒ Doppler Effect for Sound Waves
- Sound waves are mechanical waves that require a medium (like air or water). The Doppler effect in sound is affected by:
- – Speed of source
- – Speed of observer
- – Speed of sound in the medium
- Figure 1 Doppler effect for sound waves
- ⇒ Example:
- A car honking its horn moves toward you → the pitch sounds higher
- As it moves away → the pitch sounds lower
- ⇒ Mathematical Expression (Sound):
- If the source or observer is moving, the observed frequency is given by:
- [math]f’ = f \left( \frac{v + v_o}{v + v_s} \right)[/math]
- Where:
- – [math]f'[/math] = observed frequency
- – f = actual frequency of the source
- – v = speed of sound in the medium
- – [math]v_o[/math] = speed of the observer (positive if moving toward the source)
- – [math]v_s[/math] = speed of the source (positive if moving away from observer)
- ⇒ Doppler Effect for Electromagnetic Waves
- For electromagnetic (EM) waves (like light, radio, microwaves), the Doppler effect occurs even in a vacuum and is governed by relativistic principles if speeds are very high.
- Figure 2 Doppler effect for electromagnetic waves
- Applications:
- – Redshift: Light from a star moving away appears shifted toward red (lower frequency).
- – Blueshift: Light from a star moving toward us appears blue (higher frequency).
- – Used in astronomy, radar, and cosmic measurements.
- ⇒ EM Doppler Shift Formula (non-relativistic, low speeds):
- [math]f’ = f(1 \pm \frac{v}{c})[/math]
- Where:
- – f = actual frequency
- – [math]f'[/math]= observed frequency
- – v = relative speed between source and observer
- – c = speed of light
- ⇒ Relativistic Formula (at high speeds):
- [math]f’ = f \sqrt{\frac{1 + \frac{v}{c}}{1 – \frac{v}{c}}}[/math]
- This is used for high-speed objects, like spacecraft or galaxies.
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b) The Representation of the Doppler Effect in Terms of Wavefront Diagrams
- A wavefront is a line (or surface) connecting points on a wave that are in phase—like all crests.
- – In a stationary source, wavefronts are concentric circles expanding outward evenly.
- – When the source moves, wavefronts are compressed in front and spread out
- ⇒ Wavefront Diagrams
- Case 1: Stationary Source and Observer
- Wavefronts are perfect circles
- Spacing between them = wavelength
- Observer hears the actual frequency
- Figure 3 Stationary source and observer wavefront
- ⇒ Case 2: Moving Source, Stationary Observer
- Wavefronts are closer together in front of the source (shorter wavelength, higher frequency)
- Wavefronts are further apart behind (longer wavelength, lower frequency)
- ⇒ Case 3: Stationary Source, Moving Observer
- Wavefronts are still evenly spaced
- But the observer encounters them more frequently (if moving toward source)
- If observer moves toward source → hears higher frequency
- If moves away → hears lower frequency
- ⇒ Case 4: Both Source and Observer Moving
- The total Doppler effect depends on relative velocity
- Use full Doppler formula to calculate
- Figure 4 Doppler effect for two sources
- ⇒ Real-Life Applications of the Doppler Effect
| Field | Application |
|---|---|
| Police radar | Measures speed of moving cars |
| Astronomy | Determines whether galaxies are moving toward or away (redshift/blueshift) |
| Medical imaging | Doppler ultrasound measures blood flow |
| Meteorology | Doppler radar tracks weather systems |
| Sonar | Locates submarines or underwater objects |
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(c) The Doppler Effect for Light at Low Speeds ([math]v ≪ c[/math])
- When a light source and an observer move relative to each other, the wavelength and frequency of light appear shifted. This shift is called the Doppler shift, and it depends on:
- – The relative velocity v.
- – The speed of light [math]c = 3 × 10^8 m/s[/math]
- When [math]v ≪ c[/math], we can use a simplified linear approximation of the Doppler effect.
- [math]\frac{\Delta f}{f} = \frac{\Delta \lambda}{\lambda} = \frac{v}{c}[/math]
- Where:
- – f is the original (source) frequency
- – λ is the original wavelength
- – Δf is the change in frequency
- – Δλ is the change in wavelength
- – v is the relative speed between source and observer
- – c is the speed of light
- This simplified equation applies when:
- – The speed of the source or observer is much less than the speed of light
- – We’re observing small shifts in wavelength or frequency
- – The motion is along the line of sight (no angle involved)
- If the source is moving toward the observer:
- – [math]Δf > 0[/math]: Observed frequency increases (blueshift)
- – [math]Δλ < 0 [/math]: Wavelength decreases
- If the source is moving away from the observer:
- – [math]Δf < 0[/math]: Observed frequency decreases (redshift)
- – [math]Δλ > 0[/math]: Wavelength increases
- ⇒ Example
- Suppose a galaxy is moving away from Earth at [math]v = 3 × 10^6 m/s[/math](1% of the speed of light). Its original wavelength of light is λ=500nm (green light).
- Using:
- [math]\frac{\Delta \lambda}{\lambda} = \frac{v}{c} = \frac{3 \times 10^6}{3 \times 10^8} = 0.01[/math]
- So:
- [math]Δλ = 0.01 × 500 = 5nm[/math]
- New observed wavelength:
- [math]\lambda_{\text{observed}} = \lambda + \Delta \lambda \\
\lambda_{\text{observed}} = 500 + 5 \\
\lambda_{\text{observed}} = 505 \, \text{nm}[/math] - This is a redshift—the light is shifted toward red because the galaxy is receding.
| Term | Meaning |
|---|---|
| [math]\Delta f[/math] | Change in frequency due to motion |
| [math]\Delta \lambda[/math] | Change in wavelength due to motion |
| [math]v[/math] | Relative velocity between source and observer |
| [math]c[/math] | Speed of light ([math]3 × 10^8 m/s[/math]) |
| [math]\frac{\Delta f}{f} = \frac{v}{c}[/math] | Fractional frequency shift |
| [math]\frac{\Delta \lambda}{\lambda} = \frac{v}{c}[/math] | Fractional wavelength shift |
- ⇒ Real-World Applications
| Field | Use of Doppler Effect |
|---|---|
| Astronomy | Measures galaxy motion via redshift/blueshift |
| Space probes | Track probe velocity by frequency shifts |
| Radar | Detects speed of moving objects (cars, weather systems) |
| Ambulance sirens | Audible example of Doppler shift (for sound) |
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d) Shifts in Spectral Lines and the Motion of Stars and Galaxies
- Spectral lines are specific wavelengths (or frequencies) of light that are absorbed or emitted by elements in a star, galaxy, or other celestial body.
- – Each element (like hydrogen, helium, oxygen) has a unique spectral signature, like a fingerprint.
- – When light from a star is passed through a prism or diffraction grating, these lines appear in the spectrum.
- Figure 5 Shift in spectral lines and motion of stars and galaxies
- There are two main types:
- – Emission lines: Bright lines where light is emitted
- – Absorption lines: Dark lines where light is absorbed
- ⇒ When the Source Moves:
- When a star or galaxy is moving relative to Earth, the spectral lines shift due to the Doppler Effect:
- If the object is moving toward us:
- – Wavelengths get shorter
- – Spectral lines shift toward the blue end of the spectrum. This is called a blueshift
- If the object is moving away from us:
- – Wavelengths get longer
- – Spectral lines shift toward the red end of the spectrum. This is called a redshift
- The Doppler Formula for Spectral Line Shifts (Low Speed)
- If the relative velocity v is much smaller than the speed of light ccc, the fractional shift is given by:
- [math]\frac{\Delta \lambda}{\lambda} = \frac{v}{c} \quad \text{or} \quad \frac{\Delta f}{f} = \frac{v}{c}[/math]
- Where:
- – Δλ: change in observed wavelength
- – λ: rest (original) wavelength
- – v: relative speed between observer and source
- – c: speed of light
- Figure 6 Doppler shift
- ⇒ Example
- Let’s say hydrogen emits light at 3 nm (a known spectral line – H-alpha line). If this line is observed from a distant galaxy at:
- [math]\lambda_{\text{observed}} = 660.3\, \text{nm}[/math]
- Then:
- [math]Δλ = 660.3 – 656.3 = 4.0nm[/math]
- Using:
- [math]\frac{\Delta \lambda}{\lambda} = \frac{v}{c} \\
\frac{0.4}{656.3} = \frac{v}{3 \times 10^8}[/math] - So, the galaxy is moving away from us at about 1.83 million m/s.
| Observation | Interpretation |
|---|---|
| Spectral lines shifted toward red | The object is moving away (receding) |
| Spectral lines shifted toward blue | The object is moving toward us |
| No shift | The object is stationary relative to us |
- ⇒ Applications in Astronomy
- 1. Galaxy Motion & Expansion of the Universe
- – Most distant galaxies show redshift → they’re moving away
- – This led to the discovery that the universe is expanding
- – Used in Hubble’s Law:
- [math]v = H_0 . d[/math]
- Where v = recessional velocity, [math]H_o[/math] = Hubble constant, d = distance
- 2. Binary Star Systems
- – Spectral lines shift periodically due to stars orbiting each other → used to calculate orbital speeds and masses
- 3. Exoplanet Detection
- – Tiny wobbles in a star’s spectral lines can reveal orbiting planets
- 4. Rotational Speeds of Galaxies
- – One side of a spinning galaxy appears slightly blue-shifted, the other redshifted
- 5. Measuring Stellar Winds and Explosions
- – Shifts in lines help track movement of gas ejected from stars or supernovae
| Concept | Description |
|---|---|
| Spectral line | Specific wavelength of absorbed/emitted light by an element |
| Redshift | Shift toward longer wavelength; object is receding |
| Blueshift | Shift toward shorter wavelength; object is approaching |
| [math]\frac{\Delta \lambda}{\lambda} = \frac{v}{c}[/math] | Formula to calculate velocity from spectral shift |
| Applications | Galaxy movement, exoplanets, binary stars, Hubble’s law |