Welcome to the Doppler Effect!
Ever noticed how a police siren sounds high-pitched as it rushes toward you and then suddenly drops to a lower pitch as it zooms away? That’s not the driver flicking a switch—it’s Physics!
In this section, we will explore why this happens and how to use a simple formula to calculate exactly how much the pitch changes. Don't worry if it sounds a bit "fast-paced" at first; we will break it down step-by-step!
What is the Doppler Effect?
The Doppler effect is the change in the observed frequency (and wavelength) of a wave when the source of the wave is moving relative to a stationary observer.
Key Point: It is important to remember that the source (like the siren) is actually emitting the same constant frequency. It only sounds different to the person standing on the sidewalk because of the motion.
The "Slinky" Analogy
Imagine you and a friend are holding a long Slinky. If you both stand still and you wiggle your end, the coils (waves) reach your friend at a steady rate.
Now, imagine you start running toward your friend while still wiggling the Slinky. The coils will "bunch up" because you are moving into the waves you just created. Your friend sees the coils arriving much closer together (shorter wavelength, higher frequency).
If you run away, you are "stretching" the distance between each wiggle, so the coils arrive further apart (longer wavelength, lower frequency).
Quick Review:
• Source moving TOWARD you: Waves bunch up → Higher Frequency (Higher Pitch).
• Source moving AWAY from you: Waves stretch out → Lower Frequency (Lower Pitch).
Visualizing the Waves
When a source is stationary, the wave fronts are perfect circles centered on the source. Everyone around the source hears the same frequency.
When the source is moving:
1. The source "catches up" to the waves in front of it. This makes the wavelength shorter in the direction of motion.
2. The source moves away from the waves behind it. This makes the wavelength longer behind the source.
Did you know? Bats use the Doppler effect to hunt! They emit sound waves and listen to the change in frequency of the echo to figure out how fast a moth is flying away from them.
The Doppler Effect Formula
For your Cambridge AS Level exam, you only need to master the scenario where the source is moving and the observer is stationary. Here is the magic formula:
\( f_o = \frac{f_s v}{v \pm v_s} \)
Breaking down the symbols:
• \( f_o \): The observed frequency (what the person hears, measured in Hz).
• \( f_s \): The source frequency (the actual frequency of the siren, measured in Hz).
• \( v \): The speed of the sound wave (usually around \( 330 \) or \( 340 \) \( ms^{-1} \)).
• \( v_s \): The speed of the source (how fast the car/plane is moving, measured in \( ms^{-1} \)).
Choosing Plus (+) or Minus (-)
This is the part that trips most students up, but here is a simple trick to always get it right!
Recall our rule: Towards = Higher Frequency. To make the answer (\( f_o \)) a bigger number, the bottom of the fraction (the denominator) must be smaller.
1. Source Moving TOWARDS the Observer:
Use the MINUS sign.
\( f_o = \frac{f_s v}{v - v_s} \)
(Result: \( f_o \) is greater than \( f_s \))
2. Source Moving AWAY from the Observer:
Use the PLUS sign.
\( f_o = \frac{f_s v}{v + v_s} \)
(Result: \( f_o \) is smaller than \( f_s \))
Mnemonic: "Minus is for Moving towards." (Both start with M!)
Step-by-Step Problem Solving
Let's look at a typical exam-style question: A car honks its horn at a frequency of 400 Hz while driving toward a stationary student at 25 \( ms^{-1} \). The speed of sound is 340 \( ms^{-1} \). Calculate the frequency heard by the student.
Step 1: List your variables.
\( f_s = 400 \) Hz
\( v_s = 25 \) \( ms^{-1} \)
\( v = 340 \) \( ms^{-1} \)
Step 2: Decide on the sign.
The car is moving towards the student. To get a higher frequency, we need a smaller denominator. Use minus (-).
Step 3: Plug and play!
\( f_o = \frac{400 \times 340}{340 - 25} \)
\( f_o = \frac{136000}{315} \)
\( f_o \approx 432 \) Hz
Step 4: Sense check.
Is 432 Hz higher than 400 Hz? Yes! Since the car was moving toward us, this makes sense.
Common Mistakes to Avoid
• Mixing up \( v \) and \( v_s \): Always remember that \( v \) is the speed of sound (the big number, usually 340), and \( v_s \) is the speed of the car/source (the smaller number).
• Units: Make sure your speeds are in \( ms^{-1} \). If the question gives you \( km/h \), you must convert it first!
• The "Change" vs. the "Observed": Read carefully. Does the question ask for the observed frequency or the change in frequency? If it asks for the change (\( \Delta f \)), you must subtract: \( f_o - f_s \).
Key Takeaways
• The Doppler Effect is an apparent change in frequency due to relative motion.
• Moving Towards: Wavelength decreases, observed frequency increases (use \( v - v_s \)).
• Moving Away: Wavelength increases, observed frequency decreases (use \( v + v_s \)).
• The actual frequency emitted by the source does not change.