Waves

Waves

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

• A wave is a periodic disturbance that carries energy and momentum through space or a medium, while the medium’s particles only oscillate about their mean positions — no net transport of matter.
• The general progressive sinusoidal wave: y(x, t) = A sin(kx − ωt + φ), where A is amplitude, k = 2π/λ is the wave number, ω = 2πf = 2π/T is the angular frequency, and φ is the initial phase.
• Universal relations: v = fλ = ω/k. Wave speed in a stretched string: v = √(T/μ), with T = tension and μ = linear mass density.
• For sound in an ideal gas (Newton–Laplace correction): v = √(γP/ρ); in a solid rod v = √(Y/ρ); on a string v = √(T/μ).
• Standing waves: nodes at points of zero displacement, antinodes at maximum. Open pipe fundamental f = v/(2L); closed pipe fundamental f = v/(4L). Organ pipe JEE problems hinge on this.
• Beats: f_beats = |f₁ − f₂|. Doppler: f’ = f (v ± v_o)/(v ∓ v_s) — sign convention is the most-tested trap.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Wave Kinematics

A progressive wave travelling in the +x direction is described by y(x, t) = A sin(kx − ωt + φ). The argument (kx − ωt + φ) is the phase. Two points separated by integer multiples of λ share the same phase; points separated by λ/2 differ by phase π (and therefore oscillate in opposite sense). For a wave travelling in −x direction, replace −ωt with +ωt. Wave speed is determined purely by the medium: v = λ/T = ω/k, so frequency is fixed by the source while wavelength adjusts when the wave enters a new medium.

Speed in Different Media

• String: v = √(T/μ), where μ = m/L. Doubling tension increases speed by √2; quadrupling μ halves it.
• Sound in gas (Newton–Laplace): v = √(γRT/M) = √(γP/ρ). Newton’s isothermal formula √(P/ρ) gives ≈ 280 m s⁻¹ in air, while the corrected adiabatic value gives the actual ≈ 330 m s⁻¹ — a standard JEE question tests this discrepancy.
• Solid rod (longitudinal): v = √(Y/ρ); transverse wave on a rod is √(G/ρ), where G is shear modulus.

Superposition and Standing Waves

The superposition principle states the resultant displacement is the algebraic sum of individual displacements. Two identical waves travelling in opposite directions superpose to form a standing wave: y = 2A sin(kx) cos(ωt), with nodes at x = nλ/2 and antinodes at x = (2n+1)λ/4. Energy is trapped; no net propagation occurs. Resonance in a stretched string of length L with both ends fixed: only frequencies f_n = (n/2L)√(T/μ) (n = 1, 2, 3, …) are allowed — these are the harmonics, with n = 1 being the fundamental.

Interference of Sound

At a point with path difference Δx, the phase difference Δφ = (2π/λ)Δx. Constructive (loud): Δx = nλ. Destructive (quiet): Δx = (2n+1)λ/2.

Beats

Superposition of two waves of nearly equal frequencies f₁, f₂ produces amplitude modulation at f_beat = |f₁ − f₂|. The ear hears a tone whose intensity waxes and wanes at this rate. JEE restricts beats to audio range (≤ ~10 Hz perceptible). Zero beat implies equal frequencies.

Doppler Effect

The observed frequency when source and observer move relative to a stationary medium is f’ = f [(v ± v_o)/(v ∓ v_s)], with upper signs for approach. The mnemonic: numerator increases when observer moves toward source, denominator decreases when source moves toward observer — both raise pitch.

Exam Patterns

JEE Main typically tests 1 question (3% weightage) on Waves, most often: (i) standing wave / organ pipe numericals, (ii) string wave speed with mass-per-unit-length ratio, (iii) beat-counting, (iv) Doppler sign in chase or siren problems, and (v) interference / path-difference MCQs.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Subtleties

• Phase vs path difference on reflection: A wave reflected from a denser medium suffers a phase change of π (equivalent path difference λ/2); reflection from a rarer medium incurs no phase change. This single rule resolves most closed/open organ-pipe boundary-condition problems.
• Quincke’s tube and Quincke filter: variable-length sound interferometer — adjusting the sliding U-section changes path difference, giving successive maxima/minima. Distance moved between two successive minima = λ/2, between two maxima = λ. This is the experimental method JEE cites for the speed of sound.
• Apparent frequency in a moving medium (advanced): if the wind blows with velocity w toward the observer, replace v by (v + w) in the numerator only for the source’s frame. JEE Main usually assumes w = 0.
• Organ pipe end correction: antinode is not exactly at the open end but slightly outside; effective length L_eff = L + 0.6r for a circular pipe of radius r. JEE occasionally asks end-corrected fundamentals.
• Group vs phase velocity: v_p = ω/k; v_g = dω/dk. JEE Main does not require group velocity, but v_g = v_p for non-dispersive media (string, sound) is a useful sanity check.

Common Mistakes

• Mixing up the sign in the Doppler denominator — when the source moves toward the observer, write v ∓ v_s with the minus sign: f’ = f · v/(v − v_s).
• Treating wave speed as f · λ while changing medium: frequency f stays constant, but v changes, so λ must change. Forgetting this gives wrong numerical answers in refraction-of-sound and wave-on-string problems.
• Confusing displacement node (where y = 0 always) with pressure node (where pressure variation is maximum) in sound standing waves — they are spatially complementary by λ/4.
• Forgetting the 2 in y = 2A sin(kx) cos(ωt) when calculating standing-wave amplitude — peak amplitude is 2A, not A.

Worked Micro-Example

A string of length 1.2 m, mass 6 g, is stretched by a 60 N tension. The 4th harmonic is excited. Find its frequency. μ = 6 × 10⁻³ / 1.2 = 5 × 10⁻³ kg m⁻¹; v = √(60 / 5 × 10⁻³) = √12000 ≈ 109.5 m s⁻¹; λ_4 = 2L/4 = 0.6 m; f_4 = 109.5 / 0.6 ≈ 182.5 Hz.

Adjacent-Topic Links

• SHM: every particle in a sinusoidal progressive wave executes simple harmonic motion of the same frequency.
• Young’s double slit (optics): interference conditions (path difference nλ or (2n+1)λ/2) are inherited directly from wave superposition.
• Thermodynamics: speed of sound uses γ, a thermodynamic ratio — connects Waves to the kinetic theory of gases.

Practice Prompts

1. Two tuning forks of frequencies 256 Hz and 260 Hz are sounded together. How many beats are heard in 10 s, and what is the beat period?
2. A source emitting 500 Hz moves at 30 m s⁻¹ toward a stationary observer, while the wind blows at 10 m s⁻¹ from source to observer. Speed of sound = 330 m s⁻¹. Find the apparent frequency.

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