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:
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:
Doppler shift equation for a moving observer
- The ± depends on whether the observer is moving towards or away from the source
- 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
- The ± depends on whether the source is moving towards or away from the observer
- If the source is moving towards, the term in the brackets is
- If the source is moving away, the term in the brackets is
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
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
-
- Source frequency: = 3 kHz = 3000 Hz
- Observed frequency: = 2.85 kHz = 2850 Hz
- Speed of sound: = 340 m s-1
- The observer is moving away from a stationary source of sound
Step 2: write the doppler shift equation
Step 3: rearrange to find the desired quantity
Step 4: substitute in values
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:
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.