Chapter 14 — Waves

Waves are patterns of disturbance that propagate through a medium without physical transfer of matter—they transport energy and information. Mechanical waves include transverse waves (oscillations perpendicular to propagation) and longitudinal waves (oscillations parallel to propagation), with speeds determined by the medium's elastic and inertial properties.

Chapter 14 explores wave motion, starting with mechanical waves that require a material medium to propagate. It covers transverse waves on strings and longitudinal waves in fluids and solids, their mathematical description using y(x,t) = a sin(kx − ωt + φ), and key parameters: amplitude, wavelength λ = 2π/k, frequency ν = ω/2π, and wave speed v = λν. The chapter derives transverse wave speed v = √(T/μ) on strings and longitudinal wave speed v = √(B/ρ) in media, then explains superposition, interference (constructive when in-phase, destructive at phase difference π), reflection (phase reversal at rigid boundaries, no change at open boundaries), standing waves with nodes and antinodes, normal modes, and beats from frequency differences.

1. 01Mechanical waves propagate disturbances through elastic media without transferring matter; they transport energy and depend on medium properties (tension/density for strings, bulk modulus/density for fluids).
2. 02Transverse waves (oscillation ⊥ to propagation) occur in solids; longitudinal waves (oscillation ∥ to propagation) occur in all elastic media; wave speed is independent of frequency in non-dispersive media.
3. 03Sinusoidal wave equation y(x,t) = a sin(kx − ωt + φ) describes position-time evolution with amplitude a, angular wave number k = 2π/λ, angular frequency ω = 2πν, and initial phase φ.
4. 04Wave speed on a string: v = √(T/μ) where T is tension and μ is linear mass density; for sound in media: v = √(B/ρ) (bulk modulus/density) or v = √(γP/ρ) for ideal gases (Laplace correction).
5. 05Principle of superposition: net displacement = sum of individual wave displacements; produces interference (amplitudes add if in-phase, cancel if π out-of-phase), standing waves on bounded strings, and beats (frequency = |ν₁ − ν₂|).
6. 06Reflection at rigid boundaries causes π phase reversal; standing waves with fixed nodes (zero displacement) and antinodes (maximum displacement) λ/2 apart; normal modes restrict frequencies—strings fixed at both ends vibrate at νₙ = nv/(2L), pipes closed at one end at ν = (n + ½)v/(4L).