🎧 The Doppler Effect for Sound Waves: Why Sirens Change Pitch

Welcome to one of the most relatable phenomena in wave physics: the Doppler effect! Don't worry if the name sounds intimidating—you experience this concept every single day.

If you've ever heard an ambulance or a race car speeding past, you notice the pitch (the frequency) of the sound suddenly drops as it passes you. This isn't because the siren itself changes, but because of the relative motion between the sound source and you, the observer. This chapter explains exactly how we quantify that change.

Let's dive in and understand this cool concept!


1. Defining the Doppler Effect

What causes the change in frequency?

The Doppler effect describes the change in the observed frequency of a wave when the source of the wave is moving relative to the observer.

For sound waves, a change in frequency means a change in pitch.

  • When a sound source moves towards you, the pitch you hear is higher than the sound produced by the source.
  • When a sound source moves away from you, the pitch you hear is lower than the sound produced by the source.

Key Terminology:

  • Source Frequency (\(f_{\text{s}}\)): The actual frequency of the sound waves emitted by the source (e.g., the true frequency of the siren).
  • Observed Frequency (\(f_{\text{o}}\)): The frequency measured by the stationary observer.
  • Speed of Sound (\(v\)): The speed at which sound travels through the medium (usually air, about 330 m/s to 340 m/s).
  • Speed of Source (\(v_{\text{s}}\)): The speed at which the sound-emitting object is moving.

The Physical Reason: Wavefront Bunching

Imagine the source is a person clapping once every second. Each clap creates a spherical wavefront traveling outwards. When the source is stationary, these wavefronts spread out evenly, and you hear one clap per second.

But what happens if the source moves?

  1. Source Moving TOWARDS the Observer:

    As the source moves forward, it "catches up" to the waves it just emitted. The distance between consecutive wavefronts (the wavelength, \(\lambda\)) gets squeezed or compressed in the direction of motion.

    • Shorter wavelength (\(\lambda_{\text{o}}\) decreases).
    • Since \(v = f\lambda\) and \(v\) is constant (assuming the air is uniform), the observed frequency (\(f_{\text{o}}\)) must increase.
    • Result: You hear a higher pitch.
  2. Source Moving AWAY from the Observer:

    The source is pulling away from the waves it has just emitted. The wavefronts become stretched out in the direction of motion.

    • Longer wavelength (\(\lambda_{\text{o}}\) increases).
    • Since \(v = f\lambda\), the observed frequency (\(f_{\text{o}}\)) must decrease.
    • Result: You hear a lower pitch.

Key Takeaway 1: The Doppler effect is primarily a consequence of the compression or stretching of the wavelength due to the source's motion.


2. The Doppler Effect Equation (Source Moving)

For the Cambridge 9702 syllabus, you only need to apply the specific case where the source is moving and the observer is stationary.

The relationship linking the observed frequency to the source frequency is:

$$f_{\text{o}} = f_{\text{s}} \frac{v}{v \pm v_{\text{s}}}$$

Understanding the Variables:

  • \(f_{\text{o}}\) = Observed frequency (Hz)
  • \(f_{\text{s}}\) = Source frequency (Hz)
  • \(v\) = Speed of sound in the medium (\(\text{m s}^{-1}\))
  • \(v_{\text{s}}\) = Speed of the source (\(\text{m s}^{-1}\))

The critical part of this equation is choosing the correct sign (\(\pm\)) in the denominator. A small mistake here changes your entire answer!


3. Using the Sign Convention: A Simple Trick

When solving problems, you must decide whether to use \(+\) or \(-\) in the denominator. This choice depends on whether the observed frequency (\(f_{\text{o}}\)) should be higher or lower than the source frequency (\(f_{\text{s}}\)).

Goal Setting: Always look at the scenario and determine if you expect \(f_{\text{o}}\) to be larger or smaller than \(f_{\text{s}}\).

Rule 1: Source Moving TOWARDS the Observer (Higher Frequency Expected)

If the source is approaching, the sound waves are compressed, and you hear a higher frequency. Mathematically, we need the fraction \(\left(\frac{v}{v \pm v_{\text{s}}}\right)\) to be greater than 1.

To make a fraction greater than 1, you must make the denominator smaller.

Therefore, use the MINUS sign:

$$f_{\text{o}} = f_{\text{s}} \frac{v}{v - v_{\text{s}}}$$

Did you know? If the speed of the source ($v_{\text{s}}$) were equal to the speed of sound ($v$), the denominator would be zero, resulting in an infinite frequency! This is the point where the source breaks the sound barrier, creating a dramatic pressure wave known as a sonic boom.


Rule 2: Source Moving AWAY from the Observer (Lower Frequency Expected)

If the source is receding, the sound waves are stretched, and you hear a lower frequency. Mathematically, we need the fraction \(\left(\frac{v}{v \pm v_{\text{s}}}\right)\) to be less than 1.

To make a fraction less than 1, you must make the denominator larger.

Therefore, use the PLUS sign:

$$f_{\text{o}} = f_{\text{s}} \frac{v}{v + v_{\text{s}}}$$

🧠 Memory Aid for the Signs (Crucial!)

This is the most common confusion point. Remember this simple trick:

A for Away = A for Addition (use \(v + v_{\text{s}}\))
T for Towards = T for Takeaway (use \(v - v_{\text{s}}\))

Don't worry if this seems tricky at first—just remember that if the source is moving Towards you, the frequency goes Up, so you must Subtract in the denominator to make the overall fraction larger.


4. Real-World Applications of the Doppler Effect

While we study the Doppler effect using sound waves (the classic siren), this phenomenon applies to all types of waves, including electromagnetic waves (light, radar, microwaves).

  • Speed Guns (Radar): Police use radar guns that send out microwaves. When these waves reflect off a moving car, the frequency of the returning waves shifts due to the Doppler effect. Measuring this shift allows the device to calculate the car's speed.
  • Astronomy (Redshift): The Doppler effect for light waves tells us about the movement of distant stars and galaxies. If a galaxy is moving away from Earth, the light is shifted towards the red end of the spectrum (lower frequency/longer wavelength), a phenomenon called redshift.
  • Medical Imaging (Doppler Ultrasound): Used to measure blood flow. Ultrasound waves are reflected off moving red blood cells, and the change in frequency of the reflected wave allows doctors to determine the speed and direction of the blood flow.

Quick Review: Key Takeaways

Source Moving Relative to a Stationary Observer:

The observed frequency \(f_{\text{o}}\) changes based on relative speed \(v_{\text{s}}\) and the speed of sound \(v\).

General Formula: \(f_{\text{o}} = f_{\text{s}} \frac{v}{v \pm v_{\text{s}}}\)

Source Motion Expected \(f_{\text{o}}\) Sign Rule Formula Used
TOWARDS Observer Higher frequency (Higher pitch) Use MINUS (\(v - v_{\text{s}}\)) in the denominator to increase \(f_{\text{o}}\). $$f_{\text{o}} = f_{\text{s}} \frac{v}{v - v_{\text{s}}}$$
AWAY from Observer Lower frequency (Lower pitch) Use PLUS (\(v + v_{\text{s}}\)) in the denominator to decrease \(f_{\text{o}}\). $$f_{\text{o}} = f_{\text{s}} \frac{v}{v + v_{\text{s}}}$$