Sound
High-Yield Summary
- Sound is a longitudinal mechanical wave — needs a medium; speed v = √(B/ρ), so sound travels fastest in solids, slower in liquids, slowest in gases.
- Pitch = perception of frequency (not loudness). Human hearing: 20 Hz–20,000 Hz; below is infrasonic, above is ultrasonic.
- Doppler effect: f′ = f(v ± v₀)/(v ∓ vₛ) — moving toward raises perceived frequency, moving apart lowers it.
- Intensity I = P/A (W/m²) ∝ amplitude² and ∝ 1/distance²; decibel scale β = 10log(I/I₀) compresses the huge audible range logarithmically.
- Standing waves: strings and open pipes support all harmonics (λ = 2L/n); closed pipes support only odd harmonics (λ = 4L/n).
Speed of Sound
v = √(B/ρ)
- v = Speed of sound in the medium
- B = Bulk modulus — resistance to compression
- ρ = Density of the medium
- Higher B → faster sound. Higher ρ → slower sound. Solids > liquids > gases in speed of sound.
Doppler Effect
f' = f(v ± v₀)/(v ∓ vₛ)
- f' = Perceived frequency
- f = Actual source frequency
- v = Speed of sound in the medium
- v₀ = Observer's speed
- vₛ = Source's speed
- Use + in numerator / − in denominator when source and observer move toward each other; opposite signs when moving apart.
- Worked example: f = 500 Hz, v = 340 m/s, source approaches at 20 m/s → f' = 500(340/320) ≈ 531 Hz.
- Also applies to light: redshift = moving away, blueshift = moving closer.
Intensity and the Decibel Scale
I = P/A, β = 10log(I/I₀), β_f − β_i = 10log(I_f/I_i)
- I = Intensity, W/m²
- P = Power carried by the wave
- A = Area energy spreads over
- β = Sound level, in decibels (dB)
- I₀ = Reference intensity, ~10⁻¹² W/m² (hearing threshold)
- Intensity ∝ amplitude² — doubling amplitude quadruples intensity.
- Intensity ∝ 1/distance² — spreads out and weakens with distance from source.
- Worked example: intensity ×100 → Δβ = 10log(100) = 20 dB.
- Reference points: whisper ~20 dB, conversation ~60 dB, jet takeoff ~140 dB (near pain threshold).
Standing Waves in Strings and Pipes
| Boundary type | Wavelength-length relationship |
|---|---|
| String, fixed both ends (node–node) | λ = 2L/n, all integers n = 1,2,3…; first harmonic L = λ/2 |
| Open pipe, both ends open (antinode–antinode) | λ = 2L/n, all integers n = 1,2,3…; first harmonic L = λ/2 |
| Closed pipe, one end closed (node–antinode) | λ = 4L/n, odd integers only n = 1,3,5…; first harmonic L = λ/4 |
Key Terms
- Pitch
- Perception of a sound's frequency — distinct from loudness (perception of intensity).
- Shock wave / sonic boom
- Forms when an object moves at or above the speed of sound, compressing air into a high-pressure front heard as a boom.
- Attenuation
- Gradual loss of intensity/amplitude/loudness over distance due to damping effects; does not change frequency.
- Beats
- Periodic loudness fluctuation from two interfering waves of slightly different frequency; f_beat = |f₁ − f₂|.
- Ultrasound
- High-frequency sound above hearing range; a transducer sends waves that reflect at tissue boundaries to build real-time internal images.
Common MCAT Trap
- Attenuation changes amplitude/intensity/loudness but never changes frequency — a fading sound doesn't shift in pitch from attenuation alone.
- Don't confuse pitch (frequency perception) with loudness (intensity perception) — a sound can be high-pitched and quiet, or low-pitched and loud.
- Closed pipes only support odd harmonics (n = 1,3,5…) — don't apply the open-pipe/string λ = 2L/n formula to a closed pipe.
Quick Recall
Why does sound travel fastest through solids and slowest through gases?
An ambulance siren sounds higher-pitched as it approaches and lower as it passes. Why?
A closed pipe has length L. What is the wavelength of its first harmonic?