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IB DP Physics: HL復習筆記9.5.3 The Doppler Equation

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年8月1日下午6:34

The Doppler Equation

Calculating Doppler Shift

When a source of sound waves moves relative to a stationary observer, the observed frequency can be calculated using the equation below:

9-5-3-doppler-calculation-1-ib-hl

Doppler shift equation?for a moving source

The wave velocity for sound waves is 340 ms-1
The ± depends on whether the source is moving towards or away from the observer
- If the source is moving?towards?the observer, the denominator is v - us
- If the source is moving?away?from the observer, the denominator is v + us
When a source of sound waves remains stationary, but the observer is moving relative to the source, the observed frequency can be calculated using the equation below:

9-5-3-doppler-calculation-2-moving-observer-ib-hl

Doppler shift equation for a moving observer

The ± depends on whether the source is moving towards or away from the observer
- If the observer is moving?towards?the source, the numerator is?v?+?uo
- If the observer is moving?away?from the source,?the numerator is?v???uo
These equations can also be written in terms of wavelength
- For example, the equation for a moving source is shown below:

Doppler shift equation for a moving source in terms of wavelength

Worked Example

A police car siren emits a sound wave with a frequency of 450 Hz. The car is traveling away from an observer at speed of 45 m s-1. The speed of sound is 340 m s-1. Which of the following is the frequency the observer hears?

???A.?519 Hz? ? ? ? ? ? ? ?B.?483 Hz? ? ? ? ? ? ? ?C.?397 Hz? ? ? ? ? ? ? ?D.?358 Hz

WE-Doppler-shift-equation-answer-image

Worked Example

A?bank robbery has occurred and the alarm is sounding at a frequency of 3 kHz. The thief jumps into a car which accelerates and reaches a constant speed. As he drives away at a constant speed, the frequency decreases to 2.85 kHz.?The speed of sound is 340 m s-1. Determine at what speed must he be driving away from the bank.

Step 1: List the known quantities

Step 5: State final answer

- The bank robber must be driving away at a constant speed of?17 ms-1?based on the change in frequency heard

Calculating Doppler Shift of Light

Doppler shift can be calculated with relation to a light emitting source
- For example, a galaxy moving towards or away from Earth
Doppler shift for light is?complicated, however if the?speed?of the observer or source is?small (non-relativistic)?compared to the speed of light, then this equation becomes?simpler
The Doppler shift for a light-emitting non-relativistic source is described using the equation:

9-5-3-doppler-light-related-ib-hl

Doppler shift equation relating wavelength change for a moving source

Where:
- Δf = change in frequency in Hertz (Hz)
- f0?= reference frequency in Hertz (Hz)
- λ = observed wavelength of the source in metres (m)
- λ0?= reference wavelength in metres (m)
- Δλ = change in wavelength in metres (m)
- v?= velocity of a galaxy in metres per seconds (m/s)
- c?= the speed of light in?metres per second (m/s)

This means that the change in wavelength,?Δλ:

Δλ = λ – λ0

This equation can be used to calculate the velocity of a galaxy if its wavelength can be measured and compared to a reference wavelength
Since the fractions have the same units on the numerator (top number) and denominator (bottom number), the Doppler shift has?no units

Worked Example

Light emitted from a star has a wavelength of 435 × 10-9?m. A distance galaxy emits the same light but has a wavelength of 485 × 10-9?m. Calculate the speed at which the galaxy is moving relative to Earth. The speed of light = 3 × 108?m/s.

8.3.2-Doppler-Shift-Worked-Example