Sound Waves

Sound Waves

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your JAMB UTME Physics paper.

Sound is a longitudinal mechanical wave that needs a material medium; it travels as alternating compressions and rarefactions parallel to the direction of propagation. In air at 0 °C, v ≈ 330 m/s, rising by roughly 0.6 m/s for every 1 °C increase.

• Wave equation: v = fλ, where v is speed (m/s), f is frequency (Hz), λ is wavelength (m).
• Loudness ∝ amplitude²; pitch depends on frequency.
• Open pipes (both ends open) support all harmonics; closed pipes (one end closed) support only odd harmonics (1st, 3rd, 5th…).
• Audible band: 20 Hz – 20,000 Hz; below is infrasonic, above is ultrasonic.
• JAMB angle: expect 1–2 MCQs on v = fλ, pipe resonance, or string frequencies.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months before the exam.

Speed in Different Media

The speed of sound depends on the elasticity and density of the medium. In a solid rod, v = √(E/ρ); in a gas, v = √(γP/ρ). In air the convenient form is v = 330 + 0.6T, where T is temperature in °C. Solids transmit sound fastest (steel ≈ 5000 m/s), liquids are intermediate (water ≈ 1500 m/s), and gases are slowest because of low elasticity.

Loudness, Pitch and the Decibel Scale

Loudness is the perceptual response to amplitude — doubling amplitude raises intensity by a factor of four. The intensity level in decibels is L = 10 log₁₀(I/I₀), with I₀ = 10⁻¹² W/m² as the threshold of hearing. Pitch is set purely by frequency; a 512 Hz tuning fork always reads higher than a 256 Hz one, regardless of amplitude.

Property	Depends on	Typical exam question
Speed of sound	Medium & temperature	”Why does sound travel faster in steel than in air?”
Pitch	Frequency	”Raising f with constant A makes the sound…”
Loudness	Amplitude	”Doubling amplitude increases intensity by a factor of…”
Quality (timbre)	Harmonic content	”Two instruments at 440 Hz sound different because…”

Vibrating Strings and Pipes

A stretched string obeys f = (1/(2L))√(T/μ), with T = tension (N) and μ = mass per unit length (kg/m). For pipes, the end correction of 0.6r is added to the physical length so effective length Lₑ = L + 0.6r.

• Open pipe, fundamental: f₁ = v/(2Lₑ); overtones at 2f₁, 3f₁, 4f₁ …
• Closed pipe, fundamental: f₁ = v/(4Lₑ); overtones only at 3f₁, 5f₁, 7f₁ …
• Minimum echo distance ≈ 17 m, assuming v ≈ 340 m/s and a 0.1 s persistence of hearing.

Exam trap: a closed pipe does not produce a 2nd harmonic — the first overtone there is already the 3rd harmonic.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline who want mastery.

Why Only Odd Harmonics in a Closed Pipe

A closed end is a displacement node; an open end is a displacement antinode. Fitting a quarter wavelength into the pipe gives the fundamental (¼ λ). The next arrangement that places a node at the closed end and an antinode at the open end needs three quarters of a wavelength (¾ λ), forcing the second allowed wavelength to be one-third of the fundamental — hence only odd harmonics (1, 3, 5 …) survive. Open pipes accept both node–antinode and antinode–antinode boundaries, so every integer harmonic appears.

Echoes, Reverberation and Resonance

An echo is a single distinct reflection heard when the path difference exceeds about 17 m (≈ 0.1 s at 340 m/s). Reverberation is the persistence of sound from multiple overlapping reflections inside a hall; excessive reverberation muffles speech, so auditoria are lined with absorbent materials. Acoustic resonance explains the louder note when a tuning fork is held over a tube whose air column length matches a quarter wavelength of the fork’s frequency — the air column amplifies the fork’s vibration through sympathetic oscillation.

Mistake	Correct treatment
Using v = √(γP/ρ) for a metal rod	Use v = √(E/ρ) for solids
Treating pitch and loudness as interchangeable	Pitch = frequency, loudness = amplitude
Ignoring temperature correction	v in air = 330 + 0.6T (°C)
Forgetting end correction 0.6r	Effective length Lₑ = L + 0.6r
Assuming sound can travel in vacuum	Bell-in-jar experiment disproves this

Worked Micro-Example

A pipe closed at one end is 0.25 m long with radius 0.01 m. Take v = 340 m/s. Effective length Lₑ = 0.25 + 0.6(0.01) = 0.256 m. Fundamental f₁ = v/(4Lₑ) = 340/(4 × 0.256) ≈ 332 Hz. First overtone = 3f₁ ≈ 996 Hz; second overtone = 5f₁ ≈ 1660 Hz.

Practice Prompts

1. A 0.5 m open pipe resonates at 340 Hz. Find the speed of sound and the wavelength of the fundamental.
2. A string of mass per unit length 2 × 10⁻³ kg/m is stretched with 80 N tension over 0.6 m. Calculate its fundamental frequency.

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