understand that when a source of sound waves moves relative to a stationary observer, the observed frequency is different from the source frequency (understanding of the Doppler effect for a stationary source and a moving observer is not required)

Doppler Effect for Sound Waves – Cambridge International AS & A Level Physics 9702

1. Objective

Understand how the frequency heard by a stationary observer changes when the source of sound moves towards or away from the observer, and be able to apply the Doppler‑effect formula for a moving source (observer motion is optional but the full formula is given for completeness).

2. Syllabus Context

• Topic 7 Waves → 7.3 Doppler effect for sound waves (AS).
• Prerequisite concepts: longitudinal waves, wave speed, wavelength and frequency (7.1 – 7.2).
• Later links: 7.4 Superposition & interference, 7.5 Standing waves, A‑Level extensions (ultrasound imaging, astrophysics).
• This note is a stand‑alone supplement; other modules (kinematics, dynamics, electricity, etc.) are covered elsewhere.

3. Key Syllabus Skills (AO1 & AO2)

4. Basic Idea

When a source moves, the spacing between successive wave‑fronts in the direction of motion is altered while the speed of sound v in the medium stays constant.

• Source moving towards the observer – wave‑fronts are compressed → wavelength λ′ is shorter → observed frequency f′ is higher.
• Source moving away from the observer – wave‑fronts are stretched → wavelength is longer → observed frequency is lower.

5. Assumptions & Limitations

• The medium is uniform, stationary and isotropic (no wind or temperature gradients).
• Speed of sound v is constant for the given temperature and pressure (≈ 340 m s⁻¹ in dry air at 15 °C).
• Source speed u is much less than v (u ≪ v) so the source never overtakes its own waves.
• Speeds are non‑relativistic; relativistic corrections are ignored.
• If the observer also moves, the more general formula (Box 1) must be used.

6. Derivation – Moving Source, Stationary Observer

Let

• fₛ – frequency emitted by the source (Hz)
• v – speed of sound in the medium (m s⁻¹)
• u – speed of the source relative to the medium (m s⁻¹), positive when the source moves towards the observer
• λ′ – wavelength that reaches the observer (m)
• f′ – frequency heard by the observer (Hz)

When the source is at rest, the wavelength is

\[

\lambda = \frac{v}{f_s}

\]

During one period \(T = 1/f_s\) the source travels a distance \(uT\). The next crest is therefore emitted \(uT\) closer (approaching) or farther (receding) from the observer. The effective wavelength in front of a source moving towards the observer becomes

\[

\lambda' = \lambda - uT = \frac{v}{fs} - \frac{u}{fs}

= \frac{v-u}{f_s}

\]

Because the wave speed relative to the observer is still v, the observed frequency is

\[

f' = \frac{v}{\lambda'} = \frac{v}{\dfrac{v-u}{f_s}}

= \frac{v\,f_s}{v-u}

\]

For a source moving away from the observer the sign of u is reversed, giving the compact form

\[

\boxed{\,f' = \frac{v\,f_s}{\,v \mp u\,}\,}

\]

• Upper sign (–) : source approaches → frequency increases.
• Lower sign (+) : source recedes → frequency decreases.

Box 1 – Full Doppler Formula (source ± observer motion)

If the observer also moves with speed vₒ (positive when moving towards the source), the observed frequency is

\[

\boxed{\,f' = \frac{v \pm vo}{\,v \mp u\,}\;fs\,}

\]

Use “+’’ in the numerator when the observer moves towards the source and “–’’ when moving away; use “–’’ in the denominator for an approaching source and “+’’ for a receding source.

7. Low‑Speed Approximation

When \(u \ll v\) the exact expression can be linearised:

\[

\frac{\Delta f}{f_s} \approx \pm\frac{u}{v}

\qquad

\left(\Delta f = f' - f_s\right)

\]

“+’’ applies to an approaching source, “–’’ to a receding source. This approximation is handy for quick AO2 estimates.

8. Experimental Verification (Practical Skill)

1. Mount a loudspeaker on a motorised cart whose speed u can be measured (e.g., using a photogate).
2. Connect a stationary microphone to a frequency analyser (or a smartphone app).
3. Play a pure tone of known frequency fₛ (e.g., 1000 Hz).
4. Record the frequency displayed by the analyser as the cart approaches and then recedes.
5. Compare the measured values with the predictions from the Doppler formula. Discuss sources of error such as wind, temperature gradients, timing uncertainties, and the finite size of the speaker.

9. Table of Symbols

10. Worked Example

Ambulance siren: \(fₛ = 800\;\text{Hz}\), source speed \(u = 30\;\text{m s}^{-1}\), speed of sound \(v = 340\;\text{m s}^{-1}\).

1. Approaching:

   \[

   f' = \frac{340 \times 800}{340 - 30}

   = \frac{272\,000}{310}

   \approx 877\;\text{Hz}

   \]

   The pitch heard is higher than the emitted 800 Hz.

2. Receding (after passing):

   \[

   f' = \frac{340 \times 800}{340 + 30}

   = \frac{272\,000}{370}

   \approx 735\;\text{Hz}

   \]

   The pitch now drops below the source frequency.

11. Practice Questions

1. Train horn (approaching)

   \(fₛ = 1000\;\text{Hz},\; u = 20\;\text{m s}^{-1},\; v = 340\;\text{m s}^{-1}\).

   Use \(f' = \dfrac{v f_s}{v-u}\).

   Answer: \(f' \approx 1060\;\text{Hz}\).

2. Same train, after it passes (receding)

   Same data, but the source moves away.

   Use \(f' = \dfrac{v f_s}{v+u}\).

   Answer: \(f' \approx 940\;\text{Hz}\).

3. Finding the speed of sound

   A source of known frequency \(fₛ = 500\;\text{Hz}\) moves away at \(u = 15\;\text{m s}^{-1}\). The observer measures \(f' = 450\;\text{Hz}\).

   Rearrange \(f' = \dfrac{v f_s}{v+u}\) to solve for \(v\).

   \[

   v = \frac{u f'}{f_s - f'} = \frac{15 \times 450}{500-450}=135\;\text{m s}^{-1}

   \]

   (A low value indicates a cold medium or experimental error – a useful discussion point.)

4. Paper 2/5 style – solving for source speed

   A police radar gun (treated as a stationary observer) records a Doppler‑shifted frequency of \(f' = 1120\;\text{Hz}\) from a car horn that emits \(fₛ = 1000\;\text{Hz}\). The speed of sound is \(v = 340\;\text{m s}^{-1}\).

   Find the speed of the car (assume it is moving towards the observer).

   
   Solution:

   \[

   f' = \frac{v fs}{v-u}\;\;\Longrightarrow\;\;v-u = \frac{v fs}{f'}\;\;\Longrightarrow\;\;u = v - \frac{v f_s}{f'}

   \]

   \[

   u = 340 - \frac{340 \times 1000}{1120}

   = 340 - 303.6

   \approx 36.4\;\text{m s}^{-1}

   \]

   Answer: \(u \approx 36\;\text{m s}^{-1}\) (≈ 130 km h⁻¹).

5. Observer in motion (full formula)

   A listener walks towards a stationary siren at \(v_o = 5\;\text{m s}^{-1}\). The siren emits \(fₛ = 800\;\text{Hz}\) and the speed of sound is \(v = 340\;\text{m s}^{-1}\). What frequency does the listener hear?

   Use \(f' = \dfrac{v + vo}{v}\,fs\).

   \[

   f' = \frac{340 + 5}{340}\times 800 = 1.0147 \times 800 \approx 812\;\text{Hz}

   \]

   Answer: 812 Hz.

12. Summary

• The Doppler effect for sound is a change in observed frequency caused by a change in wavelength when the source moves relative to a stationary observer; the wave speed remains constant.
• For a moving source only: \[

  f' = \frac{v\,f_s}{v \mp u}

  \] (– for approaching, + for receding).

• When both source and observer move: \[

  f' = \frac{v \pm vo}{v \mp u}\,fs

  \] (signs as described in Box 1).

• Low‑speed limit: \(\displaystyle \frac{\Delta f}{f_s} \approx \pm\frac{u}{v}\).
• Key assumptions: uniform stationary medium, \(u \ll v\), non‑relativistic speeds.
• Skills required for the Cambridge exam: define the effect, state and manipulate the formulae, perform quick approximations, solve for any unknown (including \(v\) or \(u\)), and evaluate experimental set‑ups.