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IBPhysicsDPHLRevision NotesWave BehaviourDoppler EffectEquations for the Doppler Effect of Sound

Equations for the Doppler Effect of Sound (DP IB Physics)

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Katie M

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Katie M

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The Doppler Effect of Sound

  • 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 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:

9-5-3-moving-source-wavelength-ib-hl

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 minus u subscript S over v

    • If the source is moving away, the term in the brackets is 1 space plus thin space u subscript S over v

Worked Example

A police car siren emits a sound wave with a frequency of 450 Hz. The car is travelling away from an observer at a speed of 45 m s−1.

The speed of sound is 340 m s−1.

What frequency of sound does the observer hear?

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 robber jumps into a car which accelerates and reaches a constant speed.

As he drives away at a constant speed, he hears the frequency of the alarm decrease to 2.85 kHz.

Determine the speed at which the robber must be driving away from the bank.

Speed of sound = 340 m s−1

Answer:

Step 1: List the known quantities

  • Source frequency, f = 3 kHz

  • Observed frequency, f to the power of apostrophe = 2.85 kHz

  • Speed of sound, v = 340 m s−1

Step 2: Write down the Doppler shift equation

  • The observer is moving away from a stationary source of sound, so the equation to use is

f to the power of apostrophe space equals space f space open parentheses fraction numerator v space minus space u subscript o over denominator v end fraction close parentheses

Step 3: Rearrange to find the desired quantity

f to the power of apostrophe over f space equals space open parentheses fraction numerator v space minus space u subscript o over denominator v end fraction close parentheses space space space space space rightwards double arrow space space space space space fraction numerator v f to the power of apostrophe over denominator f end fraction space equals space v space minus space u subscript o

fraction numerator v f to the power of apostrophe over denominator f end fraction space plus space u subscript o equals space v space space space space space rightwards double arrow space space space space space space u subscript o space equals space v space minus space fraction numerator v f to the power of apostrophe over denominator f end fraction

u subscript o space equals space v space open parentheses 1 space minus space f to the power of apostrophe over f close parentheses

Step 4: Substitute the values into the equation

u subscript o space equals space 340 cross times open parentheses 1 space minus space fraction numerator 2.85 over denominator 3 end fraction close parentheses space equals space 17 space straight m space straight s to the power of negative 1 end exponent

  • The robber must be driving away at a constant speed of 17 m s−1 based on the change in frequency heard

Examiner Tips and Tricks

Pay careful attention as to whether you need to + or – sign in the relevant equation! If it helps, label the 'observer' and 'source' on in your question on the exam paper.

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    Katie M

    Author: Katie M

    Expertise: Physics

    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.