Lesson 12: Waves & Sound

Week 12 — Vibrations, waves, and the physics of sound

🎯 Learning Objectives

Describe wave properties (wavelength, frequency, amplitude, speed)
Understand interference and standing waves
Analyze the Doppler effect and sound intensity

🎯 Intuition First: What Is a Wave?

Think of a stadium "wave" — people stand up and sit down, but they don't move from their seats. The disturbance travels around the stadium, but the people don't. That's a wave: a disturbance that transfers energy without transferring matter.

💡 Two Types:

Transverse waves: disturbance perpendicular to direction of travel (light, waves on a string)
Longitudinal waves: disturbance parallel to direction of travel (sound, compression waves)

📋 Wave Properties

Wave speed: v = fλ (frequency × wavelength)

Frequency: f (Hz = cycles per second)

Angular frequency: ω = 2πf

Wave number: k = 2π/λ

📋 Wave Equation

y(x,t) = A sin(kx − ωt) (traveling wave)

📋 Superposition and Interference

When two waves overlap, their displacements add:

y_total = y₁ + y₂

Constructive interference: waves in phase → amplitudes add
Destructive interference: waves out of phase → amplitudes cancel

📋 Standing Waves

When two identical waves traveling in opposite directions overlap, they create a standing wave with nodes (no motion) and antinodes (max motion):

Fixed string: λₙ = 2L/n, fₙ = nv/2L (n = 1, 2, 3, ...)

n = 1 is the fundamental (first harmonic), n = 2 is the second harmonic (first overtone), etc.

📝 Worked Example: Guitar String

A guitar string (length 0.65 m, mass 3 g) is stretched under tension of 80 N. What is the fundamental frequency?

Step 1 — Linear mass density:

μ = m/L = 0.003/0.65 = 0.00462 kg/m

Step 2 — Wave speed on string:

v = √(T/μ) = √(80/0.00462) = √17,316 = 131.6 m/s

Step 3 — Fundamental frequency:

f₁ = v/2L = 131.6/(2 × 0.65) = 131.6/1.30 = 101 Hz

✅ Fundamental ≈ 101 Hz (roughly the note C)

📋 The Doppler Effect

When source and observer move relative to each other, the observed frequency shifts:

f_obs = f_s × (v ± v_obs) / (v ∓ v_s)

Sign convention:

Observer moving toward source: +v_obs (higher frequency)
Observer moving away from source: −v_obs (lower frequency)
Source moving toward observer: −v_s (higher frequency)
Source moving away from observer: +v_s (lower frequency)

💡 Everyday Example:

An ambulance siren sounds higher-pitched as it approaches and lower-pitched as it moves away. The frequency doesn't change — your ear perceives it differently due to relative motion.

📝 Worked Example: Doppler Effect

A car horn (f = 500 Hz) is mounted on a car moving toward you at 30 m/s. What frequency do you hear? (v_sound = 343 m/s)

f_obs = f_s × v / (v − v_s) = 500 × 343 / (343 − 30)

f_obs = 500 × 343/313 = 500 × 1.096 = 548 Hz

✅ You hear 548 Hz (higher than the actual 500 Hz)

📋 Sound Intensity and Decibels

Intensity: I = P/A (power per unit area, W/m²)

Decibel level: β = 10 log₁₀(I/I₀)

where I₀ = 10⁻¹² W/m² (threshold of hearing).

✅ Check Your Understanding

Q1: Can two waves cancel each other completely? When?

Click to reveal answer

Yes! When two identical waves are exactly out of phase (180° or π radians apart), their displacements are equal and opposite at every point: A + (−A) = 0. This is perfect destructive interference. Noise-canceling headphones use this principle!

Q2 (mini-problem): What is the decibel level of a sound with intensity 10⁻⁶ W/m²? (I₀ = 10⁻¹² W/m²)