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 - u_s
- If the source is moving away from the observer, the denominator is v + u_s

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 observer is moving towards or away from the source
- If the observer is moving towards the source, the numerator is v + u_o
- If the observer is moving away from the source, the numerator is v − u_o

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 $1 - \frac{u_{S}}{v}$
- If the source is moving away, the term in the brackets is $1 + \frac{u_{S}}{v}$

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

- Source frequency: $f$ = 3 kHz = 3000 Hz
- Observed frequency: $f^{'}$ = 2.85 kHz = 2850 Hz
- Speed of sound: $v$ = 340 m s^-1
- The observer is moving away from a stationary source of sound

Step 2: write the doppler shift equation

f^{'} = f (\frac{v - u_{o}}{v})

Step 3: rearrange to find the desired quantity

\frac{f^{'}}{f} = (\frac{v - u_{o}}{v})

\frac{f^{'}}{f} \times v = v - u_{o}

(\frac{f^{'}}{f} \times v) + u_{o} = v

u_{o} = v - (\frac{f^{'}}{f} \times v)

Step 4: substitute in values

u_{o} = v - (\frac{f^{'}}{f} \times v) = 340 - (\frac{2850}{3000} \times 340) = 340 - 323 = 17 m s^{- 1}

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

The Doppler Equation (DP IB Physics: HL)

Revision Note