Waves and Sound

Waves and Sound

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A wave is a disturbance that transfers energy from one point to another without net transfer of matter. Sound is a longitudinal mechanical wave that travels only through a material medium as alternating compressions (high-pressure regions) and rarefactions (low-pressure regions). Two wave families are tested: transverse (oscillation ⊥ to propagation, e.g. light, waves on a string) and longitudinal (oscillation ∥ to propagation, e.g. sound). The master relationship is v = fλ, where v is wave speed (m s⁻¹), f is frequency (Hz), and λ is wavelength (m). The period T = 1/f is the time for one complete oscillation. Pitch depends on frequency, loudness on amplitude (and intensity I = P/A), and timbre (quality) on the harmonic mix. NABTEB candidates must know the echo distance rule (≈ 17 m for one echo in air at 20 °C), the speed of sound in air ≈ 331 + 0.6T °C m s⁻¹, and that closed pipes support only odd harmonics while open pipes support all harmonics.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Wave Parameters and the Universal Wave Equation

Every travelling wave is described by five linked quantities:

Symbol	Quantity	Unit
A	Amplitude	m
λ	Wavelength	m
f	Frequency	Hz
T	Period	s
v	Wave speed	m s⁻¹

They obey v = fλ and T = 1/f. A wave moving twice as fast at fixed frequency has double the wavelength; doubling the frequency at fixed speed halves the wavelength. NABTEB almost always gives two of {v, f, λ} and asks for the third — convert cm to m before substituting.

Speed of Sound and the Medium

Sound requires matter, so it cannot travel through vacuum. Its speed in air is v = 331 + 0.6θ m s⁻¹ (θ in °C), giving ≈ 343 m s⁻¹ at 20 °C. In a solid rod of Young modulus E and density ρ, v = √(E/ρ); in a stretched string of tension T and linear mass density μ, v = √(T/μ). Sound travels faster in solids than liquids than gases because stiffness rises faster than density.

Reflection, Echo and Reverberation

Sound obeys the law of reflection (angle of incidence = angle of reflection). An echo is a distinct reflected pulse heard when the reflecting surface is at least 17 m from the observer (round-trip ≈ 0.1 s, the persistence-of-hearing threshold). Reverberation is the persistence of multiple overlapping reflections in enclosed halls, reduced by soft absorbers (curtains, foam). SONAR uses echo timing to locate underwater objects.

Stationary (Standing) Waves

When two identical waves travel in opposite directions, nodes (zero amplitude) and antinodes (maximum amplitude) form. On a string fixed at both ends, only harmonics with λₙ = 2L/n survive (n = 1, 2, 3…). In a pipe open at both ends, all harmonics exist; in a pipe closed at one end, only odd harmonics (n = 1, 3, 5…) appear because the closed end is a node.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Intensity, Decibels and the Inverse-Square Idea

Sound intensity I = P/A (W m⁻²) falls off as 1/r² for a point source. The audible threshold is I₀ = 10⁻¹² W m⁻², and the sound level in decibels is L = 10 log₁₀(I/I₀). A 10 dB rise therefore means a tenfold intensity increase, not a tenfold loudness — examiners exploit this distinction.

Doppler Effect

When source and observer move relative to each other, the observed frequency is f’ = f(v ± vₒ)/(v ∓ vₛ), where the upper signs apply when they move toward each other (apparent pitch rises) and the lower signs when they move apart (pitch drops). NABTEB usually sets the medium stationary (v = speed of sound in air) and asks whether the pitch of an approaching siren is higher or lower than the receding case.

Resonance and Natural Frequency

Every elastic system has a natural frequency f₀ at which it most readily oscillates. Resonance is the large-amplitude response produced when a periodic driving force matches f₀. A forced vibration at any other frequency gives a smaller, often out-of-phase response — do not equate the two ideas.

Common Pitfalls

• Treating the echo distance as the full distance to the wall; the wall is at half the round-trip.
• Using cm for λ in v = fλ without converting.
• Forgetting that temperature changes the speed of sound by ~0.6 m s⁻¹ per °C.
• Marking a closed pipe with an even-harmoonic node pattern — only odd harmonics fit.

Worked Micro-Example

A tuning fork of frequency 512 Hz vibrates in air at 20 °C. Find the wavelength and the distance travelled by the wave in 0.5 s. v = 331 + 0.6(20) = 343 m s⁻¹ → λ = v/f = 343/512 ≈ 0.670 m; distance = vt = 343 × 0.5 = 171.5 m.

Practice Prompts

1. A closed pipe 0.50 m long resonates at 20 °C. Calculate its fundamental frequency (v = 343 m s⁻¹; closed pipe L = λ/4 → λ = 2.0 m, f = 171.5 Hz). Name the next two allowed harmonics.
2. An observer moves toward a stationary 600 Hz source at 30 m s⁻¹. Compute the apparent frequency using the Doppler formula and state whether the pitch is higher or lower than the source’s.

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