AP Physics – Doppler Effect and Shock Waves

AP Physics – Doppler Effect and Shock Waves

AP Physics: Doppler Effect & Shock Waves – Exam-Ready Study Guide

What This Is

The Doppler Effect describes how the frequency (and wavelength) of a wave changes when the source and observer are in relative motion. On the AP exam, this appears in sound waves (e.g., sirens passing by) and light waves (e.g., redshift in astronomy). Shock waves occur when a source moves faster than the wave speed, creating a "sonic boom" (e.g., a supersonic jet). These concepts test your ability to apply wave principles to real-world scenarios, often in FRQs about sound, light, or fluid dynamics.

Real-world example: The changing pitch of a police siren as it speeds past you is the Doppler Effect in action. The "crack" of a whip or the boom from a jet breaking the sound barrier is a shock wave.

Key Terms & Concepts

• Doppler Effect: The shift in frequency (and wavelength) of a wave due to relative motion between the source and observer.
• Sound: Higher pitch when approaching, lower when receding.

• Light: "Redshift" (lower frequency) when moving away, "blueshift" (higher frequency) when approaching.

• Doppler Effect Formula (Sound): [ f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right) ]

• (f') = observed frequency (Hz)
• (f) = emitted frequency (Hz)
• (v) = wave speed (m/s, e.g., 343 m/s for sound in air)
• (v_o) = observer velocity (m/s, + if moving toward source)
• (v_s) = source velocity (m/s, – if moving toward observer)

• Sign rule: Top sign if moving toward, bottom sign if moving away.

• Mach Number: Ratio of an object’s speed to the speed of sound in the medium. [ \text{Mach} = \frac{v_{\text{object}}}{v_{\text{sound}}} ]

• Mach 1 = speed of sound (~343 m/s at sea level).

• Mach > 1 = supersonic (shock waves form).

• Shock Wave: A cone-shaped wavefront formed when a source moves faster than the wave speed (e.g., supersonic jet). Creates a sonic boom (a loud, sudden pressure change).

• Mach Cone: The 3D cone of compressed waves behind a supersonic object. The angle ((\theta)) of the cone depends on the Mach number: [ \sin \theta = \frac{v_{\text{sound}}}{v_{\text{object}}} = \frac{1}{\text{Mach}} ]

• Redshift/Blueshift (Light): Doppler Effect for electromagnetic waves.

• Redshift: Source moving away-longer wavelength (lower frequency).
• Blueshift: Source moving toward-shorter wavelength (higher frequency).

• Used in astronomy to measure galaxy speeds (e.g., Hubble’s Law).

• Wavefront: A surface over which a wave has a constant phase (e.g., crests of a ripple in water).

• Relative Motion: The Doppler Effect depends on the relative velocity between source and observer, not their absolute speeds.

• Stationary vs. Moving Source/Observer:

• If the source moves, wavelength changes (compressed ahead, stretched behind).
• If the observer moves, wave speed relative to the observer changes.

Step-by-Step: Solving Doppler Effect Problems

1. Identify the scenario:
2. Is the source moving? Observer moving? Both?

3. Is it sound (use (v = 343) m/s) or light (use (c = 3 \times 10^8) m/s)?

4. Assign signs to velocities:

5. Use the top sign in the formula if the motion is toward the other object.
6. Use the bottom sign if the motion is away.

7. Example: If the source moves toward the observer, (v_s) is subtracted in the denominator.

8. Plug into the Doppler formula:

9. For sound: (f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right))

10. For light: (\frac{\Delta \lambda}{\lambda} = \frac{v}{c}) (simplified for non-relativistic speeds).

11. Solve for the unknown:

12. If asked for frequency shift, calculate (f' - f).

13. If asked for wavelength, use (v = f \lambda) to convert.

14. Check units and reasonableness:

15. Frequencies should be positive.
16. If the source is moving faster than the wave speed, expect a shock wave (Mach > 1).

Common Mistakes

• Mistake: Mixing up the signs in the Doppler formula.

• Correction: Remember: Top sign = toward, bottom sign = away. Draw a diagram to visualize motion.

• Mistake: Assuming the Doppler Effect only applies to sound.

• Correction: It applies to all waves, including light (e.g., redshift in astronomy). The formula differs slightly for light (relativistic effects matter at high speeds).

• Mistake: Forgetting that the observer’s motion affects the relative wave speed, not the wavelength.

• Correction: If the observer moves, the wave speed relative to them changes (e.g., (v + v_o) if moving toward the source).

• Mistake: Confusing Mach number with speed.

• Correction: Mach 1 = speed of sound, not a fixed speed (varies with temperature/medium). Mach 2 = twice the speed of sound.

• Mistake: Thinking shock waves are only for sound.

• Correction: Shock waves occur for any wave when the source exceeds the wave speed (e.g., bow waves from boats, Cherenkov radiation in nuclear reactors).

AP Exam Insights

1. FRQs often test:
2. Calculating observed frequency for moving sources/observers (e.g., a train whistle or ambulance siren).
3. Explaining redshift/blueshift in astronomy (e.g., why distant galaxies appear red).

4. Sketching wavefronts for stationary vs. moving sources (e.g., concentric circles vs. compressed/stretched waves).

5. Multiple-choice traps:

6. Sign errors: The exam will test whether you pick the correct sign for (v_o) or (v_s).
7. Mach number misconceptions: Questions may ask about the angle of a shock wave or whether a sonic boom occurs (Mach > 1).

8. Light vs. sound: The Doppler formula for light is different (no medium, so no (v_o) or (v_s)—just relative velocity).

9. Tricky distinctions:

10. Doppler Effect vs. Shock Waves: The Doppler Effect is a continuous shift in frequency; shock waves are a sudden pressure change when Mach > 1.

11. Observer vs. Source Motion: Moving the observer changes the relative wave speed; moving the source changes the wavelength.

12. Lab-based questions:

13. You might be asked to design an experiment to measure the Doppler shift (e.g., using a motion sensor and sound waves).

Quick Check Questions

1. Multiple Choice: A police car with a 500 Hz siren is moving toward you at 30 m/s. What frequency do you hear? (Speed of sound = 343 m/s) A) 458 Hz B) 500 Hz C) 547 Hz D) 600 Hz Answer: C) 547 Hz. Use (f' = f \left( \frac{v}{v - v_s} \right)) since the source is moving toward you.

2. Short FRQ: A jet flies at Mach 2.5 at an altitude where the speed of sound is 300 m/s. a) What is the angle of the Mach cone formed by the shock wave? b) If the jet’s frequency is 1000 Hz, what frequency does a stationary observer hear after the jet passes? Answer: a) (\sin \theta = \frac{1}{\text{Mach}} = \frac{1}{2.5} \rightarrow \theta = 23.6°). b) After passing, the source moves away: (f' = f \left( \frac{v}{v + v_s} \right) = 1000 \left( \frac{300}{300 + 750} \right) = 286) Hz.

3. Multiple Choice: Astronomers observe a star’s light is redshifted. This means the star is: A) Moving toward Earth B) Moving away from Earth C) Stationary relative to Earth D) Emitting lower-energy light Answer: B) Moving away from Earth. Redshift = longer wavelength = source receding.

Last-Minute Cram Sheet

1. Doppler Effect formula (sound): (f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)) Top sign = toward, bottom = away.
2. Mach number: (\text{Mach} = \frac{v_{\text{object}}}{v_{\text{sound}}}); Mach > 1 = supersonic.
3. Shock wave angle: (\sin \theta = \frac{1}{\text{Mach}}).
4. Redshift = moving away; blueshift = moving toward.
5. Observer moving: Changes relative wave speed ((v \pm v_o)).
6. Source moving: Changes wavelength (compressed ahead, stretched behind).
7. Sonic boom occurs when Mach > 1.
8. Speed of sound in air-343 m/s (at 20°C).
9. Light Doppler formula (simplified): (\frac{\Delta \lambda}{\lambda} = \frac{v}{c}).
10. For sound, the medium matters (e.g., speed changes in water vs. air); for light, no medium needed.