use the expression fο = f sv / (v ± vs) for the observed frequency when a source of sound waves moves relative to a stationary observer

🎺 Doppler Effect for Sound Waves

What is the Doppler Effect?

Imagine a 🚗 speeding past you while its horn blares. The sound you hear is a bit higher in pitch than the horn’s actual tone. That change in pitch is the Doppler Effect – the shift in frequency of a wave when the source and/or observer are moving relative to each other.

The Key Formula

The observed frequency \(fo\) depends on the source frequency \(fs\), the speed of sound \(v\), and the speed of the source \(v_s\) (positive when moving toward the observer, negative when moving away):



\$fo = fs \frac{v}{v \pm v_s}\$



• Use the minus sign in the denominator when the source is moving toward the observer (frequency increases).

• Use the plus sign when the source is moving away (frequency decreases).

Why Does It Work? (Quick Derivation)

  1. Sound travels at speed \(v\). A moving source emits waves at intervals \(Ts = 1/fs\).

  2. When the source moves toward the observer, each successive wave crest is emitted from a point closer to the observer, so the distance between crests is reduced.

  3. Effective wavelength seen by the observer: \(\lambdao = \lambdas - vs Ts\).

  4. Since \(f = v/\lambda\), substitute \(\lambda_o\) to get the formula above.

Example Problems

Let’s work through a couple of quick examples.

Example 1 – Ambulance Horn

An ambulance (source frequency \(fs = 600\,\text{Hz}\)) approaches you at \(vs = 20\,\text{m/s}\). Speed of sound \(v = 340\,\text{m/s}\). Find the frequency you hear.

StepCalculation
Denominator\(v - v_s = 340 - 20 = 320\)
Observed frequency\(f_o = 600 \times \frac{340}{320} \approx 637.5\,\text{Hz}\)

Example 2 – Train Moving Away

A train emits a whistle at \(fs = 400\,\text{Hz}\) and moves away at \(vs = 15\,\text{m/s}\). What frequency do you hear?

StepCalculation
Denominator\(v + v_s = 340 + 15 = 355\)
Observed frequency\(f_o = 400 \times \frac{340}{355} \approx 383.0\,\text{Hz}\)

Exam Tips & Quick Checklist

  • Identify who is moving: Source or observer.

  • Choose the correct sign: Minus for approaching, plus for receding.

  • Check units: Convert speeds to m/s if given in km/h.

  • Use the formula: \(fo = fs \frac{v}{v \pm v_s}\).

  • Round appropriately: Keep the same significant figures as the data.

Remember: the Doppler Effect is all about relative motion – the faster the relative speed, the bigger the frequency shift! 🚀