Wave Motion and Sound

Wave Motion and Sound

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Wave — Disturbance That Propagates:

A wave is a disturbance that transfers energy from one point to another without the net transfer of the medium’s particles. Waves require a medium for propagation (mechanical waves like sound), except electromagnetic waves which propagate through vacuum.

Two fundamental types:

• Transverse waves: particle displacement perpendicular to direction of wave propagation (e.g., light, waves on a string, water surface waves)
• Longitudinal waves: particle displacement parallel to direction of wave propagation (e.g., sound in air, compression waves in springs)

Wave Parameters:

• Wavelength λ: distance between two consecutive points in phase (e.g., crest to crest)
• Frequency f: number of complete oscillations per second; SI unit: Hertz (Hz)
• Period T: time for one complete oscillation; T = 1/f
• Wave velocity v: v = fλ = λ/T (this fundamental equation connects all three)
• Amplitude A: maximum displacement from equilibrium position; determines wave intensity

Sound Waves — Longitudinal Pressure Waves:

Sound is a longitudinal wave propagating as alternating compressions (high pressure) and rarefactions (low pressure) in the medium. In air at room temperature, the speed of sound is approximately 343 m/s. The speed of sound in a gas: v = √(γP/ρ) = √(γRT/M), where γ is the adiabatic index, P is pressure, ρ is density, R is the gas constant, T is absolute temperature, and M is molar mass.

For sound in air (mostly diatomic N₂ and O₂): γ = 7/5 = 1.4, M = 0.029 kg/mol, so v ≈ 331 + 0.6T(°C) m/s (approximately 343 m/s at 20°C).

⚡ ECAT Tip: The Doppler effect describes how the observed frequency changes when there is relative motion between source and observer. The general relation is f’ = f × (v ± v_o)/(v ∓ v_s), where v is the speed of sound in the medium, v_o is the observer’s speed, and v_s is the source’s speed. Choose the upper signs for approach (relative motion that reduces the separation) and the lower signs for recession. Two special cases make the sign rule concrete: for an observer moving with a stationary source, f’ = f(v ± v_o)/v, with the plus sign when the observer approaches; for a source moving toward a stationary observer, f’ = f × v/(v − v_s), which raises the observed pitch. When both source and observer move, the two effects combine in a single expression. In every case, motion that brings source and observer closer together raises the observed frequency, while motion that increases their separation lowers it.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Wave Equation Derivation:

For a wave travelling in the +x direction with speed v, displacement y = A sin(kx - ωt + φ), where k = 2π/λ is the wave number and ω = 2πf is the angular frequency. The argument (kx - ωt) is the phase. This satisfies the wave equation: ∂²y/∂x² = (1/v²) ∂²y/∂t².

The wave speed v = ω/k = 2πf/(2π/λ) = fλ. This is always true for any wave, regardless of the medium.

Stationary (Standing) Waves:

When two waves of the same frequency and amplitude travel in opposite directions superpose, they create standing waves. For a string fixed at both ends (e.g., guitar string): the boundary conditions require nodes at both ends. Allowed wavelengths: λ_n = 2L/n, where n = 1, 2, 3, … is the harmonic number. Resonant frequencies: f_n = nv/(2L), where v = √(T/μ) is the wave speed on the string (T = tension, μ = linear mass density).

For an open pipe (both ends open — antinodes at both ends): same formula f_n = nv/(2L). For a closed pipe (one end closed — node at closed end, antinode at open end): λ_n = 4L/(2n-1), f_n = (2n-1)v/(4L). Only odd harmonics exist in a closed pipe.

Sound Intensity and Loudness:

Intensity I = Power/Area = P/(4πr²) for spherical spreading. Intensity level in decibels: β = 10 log₁₀(I/I₀), where I₀ = 10⁻¹² W/m² is the reference threshold of hearing. Each 10 dB increase corresponds to a 10-fold increase in intensity and approximately a doubling of perceived loudness. Normal conversation ≈ 60 dB; a rock concert ≈ 110 dB; jet engine ≈ 140 dB.

⚡ ECAT Tip: For beats to occur, two sound waves of slightly different frequencies f₁ and f₂ must superpose. Beat frequency f_beat = |f₁ - f₂|. This is used to tune musical instruments — adjust the tension (and hence frequency) until the beat frequency becomes zero.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Derivation of Wave Speed on a String:

Consider a string element of mass dm and length ds under tension T. For a transverse wave of small amplitude, the restoring force at the crest provides the centripetal acceleration that bends the element along the wave. An equivalent energy argument gives the same result: a pulse traveling along a string with speed v carries kinetic energy ½μv²A² per unit length (μ = linear mass density), and the potential energy stored due to stretching is also ½μv²A². Equating the elastic restoring effect to the inertia of the element yields v = √(T/μ). This result shows that wave speed depends only on the physical properties of the string — its tension and mass per unit length — not on the amplitude or frequency.

Sound in Tubes — End Corrections:

Real tubes have end corrections because the antinode at an open end actually forms slightly beyond the tube opening. Effective length L_eff = L + 0.3D for a pipe of diameter D. The same principle of effective versus physical length appears in stringed instruments, where the vibrating length includes the portion that extends over the bridge, so precise tuning accounts for the full acoustic scale length rather than the nominal one.

Ultrasound and Its Applications:

Ultrasound frequencies (> 20 kHz, above human hearing range) have important applications: (1) Medical imaging — A-scan, B-scan, and Doppler ultrasound in obstetrics and cardiology; (2) Industrial non-destructive testing — detecting flaws in welds and materials; (3) Sonar — using time-of-flight to measure ocean depth and detect submarines (v_sound in water ≈ 1500 m/s, much faster than in air).

The acoustic impedance Z = ρv determines how much sound is reflected at an interface. For ultrasound imaging, the reflection at tissue interfaces depends on the impedance difference Z₁ - Z₂. This is why ultrasound works well for soft tissues, which have similar impedances, but cannot image bone or lung clearly because the impedance mismatch is too large and almost all the sound is reflected.

⚡ ECAT Pattern: ECAT frequently tests: (1) v = fλ calculations with numerical values given for any two of v, f, λ; (2) Doppler effect problems where a source or observer moves and students must determine whether the observed frequency increases or decreases; (3) standing wave patterns in strings and pipes, identifying nodes and antinodes; and (4) intensity level in decibels calculations. A typical ECAT problem: “A source of frequency 500 Hz moves toward a stationary observer at 30 m/s. If the speed of sound is 340 m/s, what is the observed frequency?” f’ = f × v/(v - v_s) = 500 × 340/(340 - 30) = 500 × 340/310 ≈ 548 Hz.