Waves

2.1 Core concepts

• A wave is a pattern of disturbance that moves through a medium without actual physical transfer of matter as a whole; the medium oscillates while energy and information propagate (NCERT §14.1, p. 278).
• Mechanical waves (string, water, sound, seismic) need an elastic medium and arise from coupling between oscillating constituents; electromagnetic waves do not need a medium and travel at c = 299,792,458 m s⁻¹ in vacuum; matter waves are associated with electrons, protons, atoms, etc. (NCERT §14.1, p. 279).
• In a transverse wave the constituents oscillate perpendicular to the direction of propagation (e.g. a wave on a stretched string); in a longitudinal wave they oscillate along the direction of propagation (e.g. sound in a long air-filled pipe with a piston) (NCERT §14.2, pp. 280–281).
• Transverse waves require shear elasticity, so they can propagate only in solids and on the surface of liquids; longitudinal waves need only bulk/compressional elasticity and so propagate in solids, liquids and gases. In steel both can propagate; in air only longitudinal (NCERT §14.2, p. 281; Points to ponder 4, p. 297).
• A sinusoidal travelling wave on a string is described by y(x, t) = a sin(kx − ωt + φ); for a longitudinal wave the same form is used with displacement s(x, t) (NCERT §14.3, p. 281; Eqs. 14.2 and 14.9, p. 283).
• Amplitude a is the maximum displacement of a particle from equilibrium; phase (kx − ωt + φ) determines the displacement at any (x, t); φ is the initial phase angle at x = 0, t = 0 (NCERT §14.3.1, p. 282).
• Wavelength λ is the minimum distance between two points of equal phase; angular wave number k = 2π/λ with SI unit rad m⁻¹ (NCERT §14.3.2, p. 283, Eq. 14.6).
• Period T is the time for one complete oscillation of a particle, angular frequency ω = 2π/T, and frequency ν = 1/T = ω/(2π), measured in hertz (NCERT §14.3.3, p. 283, Eqs. 14.7–14.8).
• Speed of a travelling wave is v = ω/k = λ/T = νλ; in one period the wave pattern advances exactly one wavelength (NCERT §14.4, p. 284, Eqs. 14.11–14.12).
• For a transverse wave on a stretched string with tension T and linear mass density µ = m/L, the speed is v = √(T/µ); it depends only on the medium, not on wavelength or frequency (NCERT §14.4.1, p. 285, Eq. 14.14).
• For a longitudinal wave in a fluid of bulk modulus B and density ρ, v = √(B/ρ); in a linear solid bar (Young's modulus Y), v = √(Y/ρ) (NCERT §14.4.2, p. 286, Eqs. 14.19–14.20).
• Newton's formula for sound in an ideal gas, treating compressions/rarefactions as isothermal, gives v = √(P/ρ) ≈ 280 m s⁻¹ at STP, about 15% below the measured 331 m s⁻¹ (NCERT §14.4.2, pp. 286–287, Eqs. 14.22–14.23).
• Laplace correction notes the variations are adiabatic (PVγ = constant), so the adiabatic bulk modulus is B_ad = γP and v = √(γP/ρ). For air γ = 7/5, giving 331.3 m s⁻¹, matching experiment (NCERT §14.4.2, p. 287, Eq. 14.24).
• Principle of superposition: when waves overlap, the net displacement is the algebraic sum y(x, t) = y₁(x, t) + y₂(x, t) + … = Σ fᵢ(x − vt); this principle underlies interference (NCERT §14.5, pp. 287–288, Eqs. 14.25–14.26).
• Two waves of equal a, k, ω differing in phase by φ add to y = 2a cos(φ/2) sin(kx − ωt + φ/2); φ = 0 gives constructive interference (amplitude 2a), φ = π gives destructive interference (zero displacement everywhere) (NCERT §14.5, p. 288, Eqs. 14.31–14.34).
• A wave reflected at a rigid boundary undergoes a phase change of π (the boundary point must remain at zero displacement); at an open/free boundary there is no phase change (NCERT §14.6, pp. 288–289, Eqs. 14.35–14.36).
• When two identical waves travel in opposite directions, superposition gives a standing wave y(x, t) = 2a sin kx cos ωt — same ω at every point but amplitude 2a sin kx varies with x. Points of zero amplitude are nodes; of maximum amplitude are antinodes; consecutive nodes (or antinodes) are λ/2 apart (NCERT §14.6.1, pp. 289–291, Eqs. 14.37–14.39).
• For a stretched string of length L fixed at both ends (nodes at both ends), L = nλ/2, so allowed wavelengths λ = 2L/n and frequencies νₙ = nv/(2L), n = 1, 2, 3 … — all harmonics are present; n = 1 gives fundamental ν₁ = v/(2L) (NCERT §14.6.1, p. 291, Eqs. 14.40–14.42).
• For a pipe closed at one end (node at closed end, antinode at open end), L = (n + ½)λ/2, so νₙ = (n + ½) v/(2L) for n = 0, 1, 2, … — only odd harmonics ν₁ = v/4L, 3v/4L, 5v/4L, … are present (NCERT §14.6.1, pp. 291–292, Eqs. 14.43–14.44).
• A pipe open at both ends has antinodes at both ends; like the string fixed at both ends, all harmonics are present with ν₁ = v/(2L), and νₙ = nv/(2L) (NCERT §14.6.1, p. 292; Example 14.5, p. 292).
• Beats occur when two harmonic sound waves of close (but unequal) frequencies superpose. The resultant has average angular frequency ωₐ = (ω₁ + ω₂)/2 modulated by amplitude that oscillates at 2ωᵦ = ω₁ − ω₂, so the audible beat frequency is ν_beat = |ν₁ − ν₂| (NCERT §14.7, pp. 293–294, Eqs. 14.45–14.48).
• The wave function y(x, t) must be a solution of the linear wave equation ∂²y/∂t² = v² ∂²y/∂x², so any function of the form f(x − vt) describes a wave travelling in the +x direction at speed v, and g(x + vt) describes one travelling in the −x direction (NCERT §14.3, p. 281).
• For sound in an ideal gas, v depends only on temperature (not on pressure) because P/ρ is set by T via the ideal-gas law; in air v ≈ 331 m s⁻¹ at 0 °C and increases by roughly 0.61 m s⁻¹ per °C rise — a standard NCERT remark (NCERT §14.4.2, p. 287).
• A travelling wave on a string transports both kinetic energy (oscillating string elements) and elastic potential energy (string segments alternately stretched); the average power transmitted is proportional to a²ω² so doubling either amplitude or frequency quadruples the power carried — relevant to NCERT examples on string waves (NCERT §14.4, p. 285).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig. 14.1 (p. 279): chain of coupled springs showing how a disturbance at one end travels along, while each spring only oscillates about its equilibrium — the model for mechanical wave propagation.
• Fig. 14.2 / 14.3 (p. 280): pulse and sinusoidal transverse wave on a stretched string; string elements oscillate in y while the wave moves in x.
• Fig. 14.4 (p. 280): longitudinal wave generated in a piston-driven air column — compressions and rarefactions parallel to the propagation direction.
• Fig. 14.5 / 14.6 (p. 282): meaning of a, ω, k, φ in y = a sin(kx − ωt + φ), and snapshots of a harmonic wave at successive times showing the crest advancing.
• Fig. 14.7 (p. 283): displacement of one element as a function of time — period T, amplitude a.
• Fig. 14.8 (p. 284): two snapshots of a wave at t and t + Δt to read off the phase speed v = Δx/Δt.
• Fig. 14.9 (p. 287): two equal-and-opposite pulses crossing — instantaneous cancellation illustrating superposition.
• Fig. 14.10 (p. 288): resultant of two harmonic waves of equal amplitude — constructive (φ = 0, amplitude 2a) versus destructive (φ = π, zero).
• Fig. 14.11 (p. 289): reflection of a pulse from a rigid boundary with phase change π.
• Fig. 14.12 (p. 290): two oppositely-travelling waves combining into a stationary wave with fixed nodes.
• Fig. 14.13 (p. 291): first six harmonics of a string fixed at both ends — ν₁ : ν₂ : ν₃ … = 1 : 2 : 3 …
• Fig. 14.14 (p. 292): first six odd harmonics of an air column closed at one end — ν₁, 3ν₁, 5ν₁ … only.
• Fig. 14.15 (p. 293): first four harmonics of a pipe open at both ends — all harmonics present.
• Fig. 14.16 (p. 294): superposition of 11 Hz and 9 Hz waves giving a 2 Hz beat envelope.

2.4 Common confusions / NTA trap points

• Confusing wind with sound: wind carries air bodily; a sound wave only carries the compression/rarefaction pattern — no net flow of air (NCERT §14.2, p. 281; Points to ponder 1, p. 297).
• Forgetting that transverse waves cannot propagate inside fluids — fluids cannot sustain shearing stress, so in air only longitudinal sound waves exist (NCERT §14.2, p. 281).
• Misusing Newton's bare v = √(P/ρ): it gives ~280 m s⁻¹ for air at STP, off by 15%. The Laplace-corrected v = √(γP/ρ) with γ = 7/5 gives 331.3 m s⁻¹, which is what NCERT uses (NCERT §14.4.2, pp. 286–287).
• Mixing up the harmonics of a closed pipe versus an open pipe: closed pipe has only odd harmonics (ν₁, 3ν₁, 5ν₁ …) while an open pipe (or a string fixed at both ends) has all harmonics (ν₁, 2ν₁, 3ν₁ …) (NCERT §14.6.1, pp. 291–292).
• In a standing wave, every particle oscillates with the same frequency and phase between two nodes but with different amplitudes — opposite to a progressive wave where amplitude is the same but phases differ (Points to ponder 5, p. 297).
• Beat frequency is the absolute difference ν₁ − ν₂, not the average; the average sets the pitch, the difference sets the waxing rate (NCERT §14.7, p. 294, Eq. 14.48).
• A common slip is treating "+" or "−" sign inside (kx ± ωt) as cosmetic — the sign sets the direction of travel: (kx − ωt) propagates in +x, (kx + ωt) in −x (NCERT §14.3, p. 281).
• Mistaking k (angular wave number, rad m⁻¹) for the ordinary wave number 1/λ — only k = 2π/λ enters the standard NCERT wave equation y = a sin(kx − ωt).
• Forgetting that on a stretched string ν depends on length and tension/density of the wire — so changing string length (fretting a guitar) raises ν without altering v, whereas tightening the string raises v and hence ν together.
• Assuming the speed of sound in water or steel can be computed with Newton-Laplace v = √(γP/ρ): that formula is for gases only; in liquids and solids use v = √(B/ρ) and v = √(Y/ρ) (NCERT §14.4.2, p. 286).

2.5 Key formulas table