AP Physics – Superposition, Standing Waves, and Interference

AP Physics – Superposition, Standing Waves, and Interference

AP Physics: Superposition, Standing Waves, and Interference – Study Guide

What This Is

Superposition, standing waves, and interference explain how waves combine, cancel, or reinforce each other—key for understanding sound, light, and even quantum mechanics. On the AP exam, you’ll analyze wave patterns (e.g., beats, harmonics) and solve problems about wave interactions. Real-world example: Noise-canceling headphones use destructive interference to cancel out unwanted sound waves by emitting an "anti-wave" that matches the incoming wave’s amplitude but is 180° out of phase.

Key Terms & Concepts

• Superposition Principle: When two or more waves overlap, the resultant displacement at any point is the algebraic sum of the individual displacements.

• Example: Two water waves crossing each other create a temporary larger or smaller wave.

• Constructive Interference: Waves in phase (crests align with crests) add together, increasing amplitude.

• Example: Two speakers playing the same note at the same time sound louder.

• Destructive Interference: Waves out of phase (crest aligns with trough) cancel each other, reducing amplitude.

• Example: Noise-canceling headphones (as above).

• Standing Wave: A wave pattern that appears stationary, formed by two identical waves traveling in opposite directions (e.g., a plucked guitar string).

• Key features: Nodes (points of zero displacement) and antinodes (points of maximum displacement).

• Fundamental Frequency (1st Harmonic): The lowest frequency at which a standing wave forms in a medium.

• Formula: f? = v / (2L) (for a string fixed at both ends or an open pipe), where:

  • v = wave speed
  • L = length of the medium

• Harmonics (Overtones): Integer multiples of the fundamental frequency.

• Formula: f? = n·f?, where n = harmonic number (1, 2, 3, ...).

• Beats: A periodic variation in amplitude (loudness) caused by two waves of slightly different frequencies interfering.

• Formula: f_beat = |f? – f?|

• Example: Tuning a guitar by listening for beats between strings.

• Path Difference (?x): The difference in distance traveled by two waves from their sources to a point.

• Constructive interference: ?x = n? (where n = 0, 1, 2, ...)

• Destructive interference: ?x = (n + ½)?

• Double-Slit Interference (Young’s Experiment): Light passing through two slits creates an interference pattern of bright (constructive) and dark (destructive) fringes.

• Formula: d·sin? = m? (for bright fringes), where:

  • d = slit separation
  • ? = angle to the fringe
  • m = order of the fringe (0, ±1, ±2, ...)
  • ? = wavelength

• Resonance: When a system is driven at its natural frequency, leading to large-amplitude oscillations.

• Example: A singer shattering a glass by matching its resonant frequency.

Step-by-Step / Process Flow

How to solve a standing wave problem (e.g., string or pipe):
1. Identify the medium and boundary conditions: - String fixed at both ends or pipe open/closed at ends-determines allowed harmonics.
2. Draw the standing wave pattern: - Label nodes (N) and antinodes (A). Count the number of loops (n = harmonic number).
3. Use the correct formula for frequency: - String fixed at both ends or open pipe: f? = n·v / (2L) - Pipe closed at one end: f? = n·v / (4L) (only odd harmonics: n = 1, 3, 5, ...)
4. Relate wavelength to length: - String/open pipe: = 2L / n - Closed pipe: = 4L / n
5. Solve for the unknown (frequency, length, speed, etc.).

How to solve an interference problem (e.g., double-slit):
1. Find the path difference (?x): - For a point on a screen, ?x = d·sin? (where d = slit separation).
2. Determine if interference is constructive or destructive: - Constructive: ?x = m? - Destructive: ?x = (m + ½)?
3. Solve for the unknown (e.g., wavelength, fringe spacing). - For small angles, sin?-tan? = y / D, where y = fringe distance from center, D = distance to screen.

Common Mistakes

• Mistake: Forgetting that closed pipes only have odd harmonics (1st, 3rd, 5th, ...).

• Correction: A pipe closed at one end has a node at the closed end and an antinode at the open end, so only odd harmonics fit. Why? Even harmonics would require a node at both ends (impossible for a closed pipe).

• Mistake: Mixing up wavelength formulas for strings vs. pipes.

• Correction:

  • String/open pipe: = 2L / n
  • Closed pipe: = 4L / n (only odd n). Why? A closed pipe’s fundamental has a wavelength 4× its length (not 2×).

• Mistake: Assuming all standing waves have the same speed.

• Correction: Wave speed (v) depends on the medium (e.g., tension in a string, temperature in air). Why? v = f·?, but f and ? change with harmonics—v stays constant for a given medium.

• Mistake: Confusing beats with harmonics.

• Correction:

  • Beats: Two waves of slightly different frequencies-amplitude varies.
  • Harmonics: Integer multiples of a single frequency-standing wave patterns. Why? Beats are a temporal effect (heard over time), while harmonics are spatial (seen in wave patterns).

• Mistake: Misapplying path difference in interference problems.

• Correction: Path difference is not the distance between slits—it’s the extra distance one wave travels to reach a point. Why? Interference depends on the phase difference caused by path length differences.

AP Exam Insights

1. FRQ Hot Topics:
2. Standing waves in pipes/strings: You’ll likely be asked to:
   • Draw the 3rd harmonic for a pipe open at both ends.
   • Calculate the fundamental frequency given length and wave speed.
   • Explain why a closed pipe has a lower pitch than an open pipe of the same length.

3. Double-slit interference: Expect questions about:

   • Fringe spacing (y = m?D / d).
   • How changing d or ? affects the pattern.

4. Multiple-Choice Traps:

5. Closed vs. open pipes: The AP exam loves testing whether a pipe is open or closed. Closed pipes only have odd harmonics!
6. Phase shifts: A wave reflecting off a fixed end (e.g., string tied to a wall) inverts (180° phase shift), but a wave reflecting off a free end (e.g., open pipe) does not.

7. Units: Ensure L (length) is in meters, not centimeters, when using v = f·?.

8. Tricky Distinctions:

9. Interference vs. diffraction: Interference = waves from two sources overlapping; diffraction = waves bending around one obstacle/slit.

10. Transverse vs. longitudinal standing waves:

    • Transverse (e.g., string): Displacement is perpendicular to wave direction.
    • Longitudinal (e.g., sound in a pipe): Displacement is parallel to wave direction (compressions/rarefactions).

11. Lab-Based Questions:

12. You might be asked to design an experiment to measure the speed of sound using resonance in a tube (e.g., adjusting water level to find harmonics).

Quick Check Questions

1. Multiple Choice: A string fixed at both ends vibrates in its 3rd harmonic. If the string length is 1.2 m and the wave speed is 120 m/s, what is the frequency of the 3rd harmonic? (A) 50 Hz (B) 100 Hz (C) 150 Hz (D) 300 Hz

Answer: (C) 150 Hz. Explanation: For a string fixed at both ends, f? = n·v / (2L). Here, n = 3, v = 120 m/s, L = 1.2 m-f? = 3·120 / (2·1.2) = 150 Hz.

1. Short FRQ: A pipe open at both ends has a fundamental frequency of 200 Hz. If the pipe is now closed at one end, what is the new fundamental frequency? Assume the speed of sound is 340 m/s.

Answer: 100 Hz. Explanation: For an open pipe, f? = v / (2L). For a closed pipe, f? = v / (4L). Closing one end halves the frequency (since L is the same but the wavelength doubles).

1. Multiple Choice: In a double-slit experiment, the 2nd-order bright fringe (m = 2) is observed at an angle of 30°. If the slit separation is 5.0 × 10 m, what is the wavelength of the light? (A) 250 nm (B) 500 nm (C) 625 nm (D) 1250 nm

Answer: (D) 1250 nm. Explanation: Use d·sin? = m?. Here, d = 5.0 × 10 m, ? = 30°, m = 2-? = (5.0 × 10)·sin(30°) / 2 = 1.25 × 10 m = 1250 nm.

Last-Minute Cram Sheet

1. Superposition: Waves add algebraically (crest + crest = bigger crest; crest + trough = cancel).
2. Standing waves: Nodes (no motion) and antinodes (max motion) form from two identical waves moving opposite directions.
3. String/open pipe harmonics: f? = n·v / (2L); = 2L / n (all harmonics allowed).
4. Closed pipe harmonics: f? = n·v / (4L); = 4L / n (only odd n).
5. Beats: f_beat = |f? – f?| (hear "wah-wah" sound).
6. Double-slit interference: d·sin? = m? (bright fringes); d·sin? = (m + ½)? (dark fringes).
7. Path difference: Constructive = n?; destructive = (n + ½)?.
8. Resonance: System driven at natural frequency-large amplitude (e.g., Tacoma Narrows Bridge collapse).
9. Closed pipes: Only odd harmonics (1st, 3rd, 5th, ...)—don’t forget this!
10. Units: Always convert L to meters and ? to meters (AP expects SI units).