Sound Waves and Doppler Effect

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Sound is a longitudinal mechanical wave that propagates through a medium by alternating compressions (regions of high pressure) and rarefactions (regions of low pressure). Because it is mechanical, it cannot travel through a vacuum — a bell rung inside an evacuated jar produces no audible signal.

Two relationships carry most of the marks. The wave equation: v = fλ, where v is wave speed in m s⁻¹, f is frequency in hertz, and λ is wavelength in metres. The Doppler equation: f′ = f(v ± vₒ)/(v ± vₛ), where v is the speed of sound in the medium, vₒ is the observer’s speed and vₛ is the source’s speed (positive when motion is toward the other party, negative when moving away).

NECO SSCE high-yield points: (1) speed of sound in air rises roughly 0.6 m s⁻¹ per °C from 331 m s⁻¹ at 0 °C; (2) pitch is determined by frequency, loudness by amplitude; (3) echoes need a minimum reflecting distance of about 17 m because the ear retains sound for ~0.1 s.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Nature and Properties of Sound

Sound waves are longitudinal, meaning particle vibration is parallel to the direction of energy transfer. A vibrating source (e.g. a tuning fork prong) pushes adjacent air molecules, creating a compression; as the prong returns, a rarefaction follows. Wavelength λ is the distance between successive compressions.

The audible range for humans is roughly 20 Hz to 20 000 Hz. Below 20 Hz lies infrasound; above 20 kHz, ultrasound. Ultrasound is used in medical imaging and SONAR because shorter wavelengths resolve smaller features.

Speed of Sound

The speed of sound depends on the medium’s elasticity and density. In air at 0 °C, v ≈ 331 m s⁻¹. The empirical temperature relation is:

v = 331 + 0.6 T (T in °C, v in m s⁻¹)

Sound travels faster in solids than in liquids, and faster in liquids than in gases, because solids have higher bulk modulus.

Echoes and Reverberation

An echo is a single distinct reflection. The distance to a reflecting wall is given by d = vt/2, where t is the measured time interval. A distinct echo requires t ≥ 0.1 s, so d ≥ (331 × 0.1)/2 ≈ 16.6 m ≈ 17 m.

Reverberation is the persistence of sound caused by many overlapping reflections in a hall. Excessive reverberation muddles speech; auditoria are designed with absorbent materials to control it.

The Doppler Effect

When source or observer moves, the observed frequency changes even though the emitted frequency stays constant. Approaching motion shortens wavefront spacing (higher f′); receding motion lengthens it (lower f′).

Sign convention for f′ = f(v ± vₒ)/(v ± vs):

Numerator: +vₒ if observer moves toward source, −vₒ if away.
Denominator: −vₛ if source moves toward observer, +vₛ if away.

Typical NECO Question Patterns

Multiple-choice on properties of longitudinal waves.
Calculate d given v and t for an echo.
Apply Doppler formula with correct sign selection when both source and observer move.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Worked Example — Echo Distance

A girl claps 60 m from a cliff. Using v = 340 m s⁻¹ in air, the time to hear the echo is t = 2d/v = (2 × 60)/340 ≈ 0.353 s. To hear a distinct echo, she must be at least d = v × 0.1/2 ≈ 17 m from the cliff.

Worked Example — Doppler Formula

A car horn emits f = 500 Hz. The car moves toward a stationary observer at 30 m s⁻¹. Take v = 330 m s⁻¹.

f′ = f × v/(v − vₛ) = 500 × 330/(330 − 30) = 500 × 330/300 = 550 Hz.

If the car then recedes at the same speed, f′ = 500 × 330/(330 + 30) = 458 Hz. The pitch drop as the car passes is the classic NECO “ambulance” scenario.

Edge Cases and Common Traps

Both moving. Write the full expression carefully: a cyclist riding toward a moving ambulance uses +vₒ in the numerator (observer toward source) and −vₛ in the denominator (source toward observer). Mixing signs is the single largest source of lost marks.
Source at the speed of sound. A source moving at v produces vₛ = v, so the denominator becomes zero — the wavefronts pile into a shock wave (sonic boom). NECO questions sometimes test this limit qualitatively.
Sound through steel vs air. Resist the “denser = faster” assumption. Steel is denser than air yet transmits sound roughly fifteen times faster because its elastic modulus dominates.
Temperature dependence. At NECO, expected answers use 330–360 m s⁻¹; if a numerical problem gives a temperature, apply v = 331 + 0.6 T.

Connections to Other Topics

Connects to wave phenomena (reflection underpinning echo formation).
Underpins acoustics in buildings and SONAR/radar (electromagnetic analog of Doppler).
Links to simple harmonic motion: a tuning fork prong executes SHM at frequency f, the source of the pressure wave.

Practice Prompts

A ship sends a sound signal toward a cliff. The echo returns in 4 s. If the speed of sound in seawater is 1500 m s⁻¹, find the distance to the cliff.
A source emitting 800 Hz moves away from a stationary observer at 20 m s⁻¹. Taking v = 340 m s⁻¹, calculate the observed frequency.

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Sound Waves and Doppler Effect