IB Physics How to Solve: IB Physics – Waves & Oscillations (Standing Waves, Doppler, Interference)

How to Solve: IB Physics – Waves & Oscillations (Standing Waves, Doppler, Interference)

Complete Guide

Introduction

"Master standing waves, Doppler shifts, and interference, and you’ll unlock 15–20% of your IB Physics Paper 2—plus real-world applications like ultrasound imaging, musical instruments, and even how bats hunt in the dark."

WHAT YOU NEED TO KNOW FIRST

1. Wave basics: Amplitude, wavelength, frequency, speed, and the wave equation v = fλ.
2. Superposition principle: When two waves meet, their displacements add algebraically.
3. Phase difference: Measured in degrees or radians; determines constructive/destructive interference.

KEY TERMS & FORMULAS

Standing Waves

Formulas: 1. Standing wave on a string (fixed ends):
L = n(λ/2) (n = 1, 2, 3…)
- L = length of string (m)
- λ = wavelength (m)
- n = harmonic number (1 = fundamental, 2 = 1st overtone, etc.)
- MEMORISE THIS

1. Frequency of nth harmonic:
   fₙ = n(v/2L)
2. fₙ = frequency of nth harmonic (Hz)
3. v = wave speed on string (m/s)
4. L = length of string (m)

5. MEMORISE THIS

6. Wave speed on a string:
   v = √(T/μ)

7. T = tension in string (N)
8. μ = linear mass density (kg/m) (μ = mass/length)
9. MEMORISE THIS

Doppler Effect

Formula: f′ = f(v ± vₒ)/(v ∓ vₛ) - f′ = observed frequency (Hz) - f = emitted frequency (Hz) - v = speed of wave in medium (m/s) (e.g., 340 m/s for sound in air) - vₒ = speed of observer (m/s) (+ if moving toward source, – if moving away) - vₛ = speed of source (m/s) (– if moving toward observer, + if moving away) - GIVEN ON EXAM SHEET (but you must know how to apply signs!)

Interference

Formulas: 1. Path difference for constructive interference:
Δx = nλ (n = 0, 1, 2…)
- MEMORISE THIS

1. Path difference for destructive interference:
   Δx = (n + ½)λ (n = 0, 1, 2…)

2. MEMORISE THIS

3. Double-slit interference (maxima):
   d sinθ = nλ

4. d = slit separation (m)
5. θ = angle to nth maximum
6. n = order of maximum (0, 1, 2…)

7. GIVEN ON EXAM SHEET

8. Double-slit interference (minima):
   d sinθ = (n + ½)λ

9. GIVEN ON EXAM SHEET

STEP-BY-STEP METHOD

Standing Waves (Fixed Ends)

1. Identify the harmonic number (n):
2. Count the number of loops (half-wavelengths) in the diagram.

3. Example: 1 loop = n=1 (fundamental), 2 loops = n=2 (1st overtone).

4. Write the standing wave equation:

5. L = n(λ/2)

6. Solve for wavelength (λ):

7. Rearrange: λ = 2L/n

8. Find frequency (f) using wave speed (v):

9. f = v/λ (or use fₙ = n(v/2L) for harmonics)

10. Check units and reasonableness:

11. Higher harmonics → shorter λ → higher f.

Doppler Effect

1. Draw a diagram:
2. Label source (S), observer (O), and their velocities.

3. Use arrows to show direction of motion.

4. Assign signs to vₒ and vₛ:

5. Observer moving toward source: +vₒ
6. Observer moving away: –vₒ
7. Source moving toward observer: –vₛ (reduces denominator)

8. Source moving away: +vₛ (increases denominator)

9. Plug into Doppler formula:

10. f′ = f(v ± vₒ)/(v ∓ vₛ)

11. Simplify and solve for f′.

12. Check if answer makes sense:

13. Moving toward → higher f (blueshift).
14. Moving away → lower f (redshift).

Interference (Double-Slit)

1. Identify given values:

2. Slit separation (d), wavelength (λ), order (n), distance to screen (D), fringe spacing (Δy).

3. Determine if it’s a maximum or minimum:

4. Maxima: d sinθ = nλ

5. Minima: d sinθ = (n + ½)λ

6. For small angles (θ < 10°), use:

7. sinθ ≈ tanθ = Δy/D

8. So, d(Δy/D) = nλ → Δy = nλD/d

9. Solve for unknown (Δy, d, λ, etc.).

10. Check units and reasonableness:

11. Larger d → smaller fringe spacing (Δy).
12. Larger λ → larger fringe spacing.

WORKED EXAMPLES

Example 1 – Standing Waves (Basic)

A guitar string of length 0.65 m vibrates at its 3rd harmonic. The wave speed on the string is 260 m/s. Calculate the wavelength and frequency of this harmonic.

Step-by-Step: 1. Harmonic number (n): 3rd harmonic → n = 3. 2. Standing wave equation: L = n(λ/2)
- 0.65 = 3(λ/2) 3. Solve for λ:
- λ = (2 × 0.65)/3 = 0.433 m 4. Find frequency (f):
- f = v/λ = 260/0.433 = 600 Hz
(or use fₙ = n(v/2L) = 3(260/(2×0.65)) = 600 Hz)

What we did and why: - Used L = n(λ/2) to find λ because the string has fixed ends. - Used f = v/λ to find frequency because wave speed is constant.

Example 2 – Doppler Effect (Medium)

A police car emits a siren at 800 Hz while moving toward a stationary observer at 30 m/s. The speed of sound is 340 m/s. What frequency does the observer hear?

Step-by-Step: 1. Diagram:
- Source (S) moving toward observer (O) at 30 m/s.
- Observer is stationary → vₒ = 0. 2. Doppler formula:
- f′ = f(v)/(v – vₛ) (source moving toward → –vₛ)
- f′ = 800(340)/(340 – 30) 3. Calculate:
- f′ = 800 × 340 / 310 = 877 Hz

What we did and why: - Used –vₛ because the source is moving toward the observer (reduces denominator). - Check: Moving toward → higher frequency (877 Hz > 800 Hz).

Example 3 – Interference (Exam-Style)

In a double-slit experiment, light of wavelength 500 nm passes through slits separated by 0.2 mm. The screen is 1.5 m away. Calculate the distance between the 2nd and 3rd bright fringes.

Step-by-Step: 1. Identify:
- λ = 500 nm = 5 × 10⁻⁷ m
- d = 0.2 mm = 2 × 10⁻⁴ m
- D = 1.5 m
- n = 2 and 3 (bright fringes) 2. Fringe spacing formula:
- Δy = nλD/d 3. Calculate y₂ and y₃:
- y₂ = 2 × 5 × 10⁻⁷ × 1.5 / 2 × 10⁻⁴ = 0.0075 m
- y₃ = 3 × 5 × 10⁻⁷ × 1.5 / 2 × 10⁻⁴ = 0.01125 m 4. Distance between fringes:
- Δy = y₃ – y₂ = 0.01125 – 0.0075 = 0.00375 m (3.75 mm)

What we did and why: - Used Δy = nλD/d because we’re dealing with small angles (θ < 10°). - Calculated positions of n=2 and n=3 separately, then found the difference.

COMMON MISTAKES

1. MISTAKE: Forgetting to convert units (e.g., mm → m, nm → m).
   WHY IT HAPPENS: Students rush and overlook unit prefixes.
   CORRECT APPROACH: Always convert to meters before plugging into formulas.

2. MISTAKE: Mixing up signs in the Doppler formula.
   WHY IT HAPPENS: Confusion about whether to add or subtract vₒ/vₛ.
   CORRECT APPROACH: Draw a diagram and label directions. Toward = +vₒ or –vₛ.

3. MISTAKE: Using the wrong harmonic number (n).
   WHY IT HAPPENS: Misidentifying loops in a standing wave diagram.
   CORRECT APPROACH: Count half-wavelengths (loops) to find n.

4. MISTAKE: Assuming all interference is constructive.
   WHY IT HAPPENS: Forgetting that path difference determines interference type.
   CORRECT APPROACH: Check if Δx = nλ (constructive) or (n + ½)λ (destructive).

5. MISTAKE: Ignoring the wave speed formula for standing waves.
   WHY IT HAPPENS: Trying to solve for frequency without knowing v = √(T/μ).
   CORRECT APPROACH: If tension or mass density is given, calculate v first.

EXAM TRAPS

1. TRAP: Doppler effect with both source and observer moving.
   HOW TO SPOT IT: Question mentions "car moving toward a moving observer."
   HOW TO AVOID IT: Assign signs separately to vₒ and vₛ. Draw a diagram!

2. TRAP: Standing waves in open pipes (not fixed ends).
   HOW TO SPOT IT: Question says "open at both ends" or shows antinodes at both ends.
   HOW TO AVOID IT: Use L = n(λ/2) for both ends open (same as fixed ends), but L = (2n + 1)(λ/4) for one end open.

3. TRAP: Interference with white light (multiple wavelengths).
   HOW TO SPOT IT: Question mentions "colored fringes" or "central white fringe."
   HOW TO AVOID IT: Remember:

4. Central fringe (n=0) is white (all λ constructively interfere).
5. Higher-order fringes show spectral separation (red farther out than blue).

1-MINUTE RECAP

"Here’s your 60-second survival guide for Waves & Oscillations: 1. Standing waves: Count loops → n → L = n(λ/2) → f = n(v/2L). Memorize the harmonic formula! 2. Doppler effect: Draw a diagram, assign signs (+ for toward, – for away), plug into f′ = f(v ± vₒ)/(v ∓ vₛ). 3. Interference: Path difference = nλ (constructive) or (n + ½)λ (destructive). For double-slit, use d sinθ = nλ or Δy = nλD/d. 4. Units: Always convert to meters. Always. 5. Check your answer: Higher harmonic = higher frequency. Moving toward = higher pitch. Larger slit separation = smaller fringe spacing. You’ve got this. Now go ace that exam!