Introduction to Wave Speed and the Doppler Effect

Have you ever stood by the side of a road and noticed how the pitch of an ambulance siren seems to drop as it zooms past you? Or have you wondered how scientists calculate exactly how fast a wave is moving across the ocean? In this chapter, we will explore the mathematical "Golden Rules" of waves and uncover the mystery behind that changing siren sound, known as the Doppler effect.

Waves are everywhere, from the light that allows you to see this page to the sound of your favorite music. Understanding how fast they travel and how their frequency changes is essential for everything from weather forecasting to medical imaging.

1. Frequency and the Time Period

Before we can calculate the speed of a wave, we need to understand two fundamental "heartbeats" of any wave: Frequency and Period.

What is Frequency?

Frequency (\(f\)) is the number of complete waves that pass a certain point every second. It is measured in Hertz (\(Hz\)).
Example: If a wave has a frequency of \(10 Hz\), it means 10 full waves are passing by every single second.

What is the Time Period?

The Time Period (\(T\)) is the time it takes for one complete wave to pass a point. It is measured in seconds (\(s\)).

The Relationship Between \(f\) and \(T\)

Frequency and Time Period are the "inverses" of each other. If you know one, you can always find the other using this formula:

\(f = \frac{1}{T}\)

Quick Tip: If a wave takes a long time to pass (large \(T\)), the frequency must be low. If it passes very quickly (small \(T\)), the frequency must be high!

Key Takeaway: Frequency is "how many per second," and Time Period is "how many seconds per wave."

2. The Wave Equation

The most important formula in this chapter links the speed of a wave to its frequency and wavelength. You will need to memorize this for your exams.

The Formula

Wave speed = frequency \(\times\) wavelength

\(v = f \times \lambda\)

  • \(v\) is the wave speed, measured in metres per second (\(m/s\)).
  • \(f\) is the frequency, measured in Hertz (\(Hz\)).
  • \(\lambda\) (the Greek letter lambda) is the wavelength, measured in metres (\(m\)).

Using the Formula Triangle

If you find rearranging equations tricky, remember this triangle: Put \(v\) at the top, and \(f\) and \(\lambda\) at the bottom.
• To find \(v\): \(f \times \lambda\)
• To find \(f\): \(\frac{v}{\lambda}\)
• To find \(\lambda\): \(\frac{v}{f}\)

Common Mistake Alert! Always check your units. If the wavelength is given in centimetres (\(cm\)), you must divide by 100 to convert it into metres (\(m\)) before using the formula.

Key Takeaway: For any wave, if you increase the frequency, the wavelength must decrease to keep the speed the same (assuming the medium doesn't change).

3. The Doppler Effect

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

Why does it happen?

Imagine a stationary toy duck bobbing in a bath. It creates circular ripples that spread out evenly in all directions. The wavelength is the same everywhere.

Now, imagine the duck starts swimming toward the right:
1. As it moves, it "catches up" with the waves it just sent forward. This bunches the waves together in front of the duck.
2. Behind the duck, the waves are left behind and stretch out.

What the Observer Sees (or Hears):

• Source moving TOWARDS you: The waves reach you more frequently. The wavelength decreases and the frequency increases. In sound, this makes the pitch sound higher.
• Source moving AWAY from you: The waves are stretched out. The wavelength increases and the frequency decreases. In sound, this makes the pitch sound lower.

Important Note: The source (like the ambulance siren) is actually emitting the same constant frequency the whole time. The change is only in what the observer detects because of the relative motion.

Did you know? The Doppler effect doesn't just happen with sound! It happens with light too. When galaxies move away from us, their light waves stretch out and look "redder"—this is known as Red-shift (which you will study in the Astrophysics section).

4. Summary Checklist

Don't worry if this seems like a lot to take in! Here is a quick summary of what you need to know for your Edexcel IGCSE:

  • Definitions: Know that frequency is waves per second and period is the time for one wave.
  • Equation 1: Be comfortable using \(f = \frac{1}{T}\).
  • Equation 2: Memorize and practice \(v = f \times \lambda\).
  • The Doppler Effect: Be able to explain that waves "bunch up" in front of a moving source (higher \(f\), shorter \(\lambda\)) and "spread out" behind it (lower \(f\), longer \(\lambda\)).
  • Energy Transfer: Always remember that waves transfer energy and information without transferring matter.

Quick Review Question: If a wave has a frequency of \(50 Hz\) and a wavelength of \(2 m\), what is its speed?
(Answer: \(v = 50 \times 2 = 100 m/s\))