How to Calculate the Doppler Effect

Finds the frequency heard when the source or the listener is moving. Closing in packs the waves together and raises the pitch; moving apart stretches them and lowers it. Enter speeds as positive when they close the gap.

When a source or a listener is moving, the frequency heard differs from the frequency emitted. This is the Doppler effect.

f' = f\dfrac{V + v_o}{V - v_s}

$f$ is the frequency of the source, $V$ the speed of sound, $v_o$ the speed of the listener and $v_s$ the speed of the source. Both speeds count as positive when they close the gap.

Why it happens

A source moving towards you travels forward in the time between emitting one wave and the next, so the waves arrive packed closer together. A shorter wavelength at the same speed of sound means a higher frequency. Moving away stretches them out and lowers it. An ambulance drops in pitch as it passes because it switches from approaching to receding.

Example

A source of 800 Hz closes at 20 m/s. Taking the speed of sound as 340 m/s, $f' = 800 \times 340 \div (340 - 20) = 850$ Hz, which is 50 Hz higher. Receding at the same speed gives $800 \times 340 \div 360 = 756$ Hz, 44 Hz lower. Note that the shift is not the same size in each direction.

Notes

When the source reaches the speed of sound the denominator becomes 0 and the formula fails. The waves can no longer run ahead and pile up into a shock wave.

A moving listener and a moving source enter the formula in different places. The same speed does not give quite the same result.