Generation, propagation, and reception of mechanical waves in fluids and solids. Acoustic wave equation, impedance, plane/spherical/standing waves, ultrasonics, room acoustics, nonlinear acoustics.
Acoustics
Acoustic wave equation
Linearised fluid wave equation ∂²_t p = c² ∇²p for small pressure perturbations in an inviscid compressible fluid at rest.
Acoustic impedance Z = ρ c
Resistance a medium offers to acoustic flow; plane-wave Z = ρc. Mismatch at boundaries produces reflection R = (Z_2 − Z_1)/(Z_2 + Z_1).
Plane acoustic wave
Solution p(x, t) = A cos(kx − ωt) of the acoustic wave equation with uniform phase over planes perpendicular to k̂.
Spherical acoustic wave
Outgoing solution p(r,t) = (A/r) f(r/c − t): intensity falls as 1/r², amplitude as 1/r. Far-field of any localised source.
Standing acoustic wave
Superposition of equal counter-propagating waves; fixed nodes/antinodes; forms modes of bounded resonators (organ pipes, strings, rooms).
Ultrasound
Acoustic waves above 20 kHz; used in medical imaging, non-destructive testing, SONAR, and therapeutic ablation.
Nonlinear acoustics
Regime where finite amplitudes invalidate linearisation; described by the Burgers or KZK equations — predicts wave steepening and shock…
Room acoustics
Modelling of sound in enclosures: Sabine reverberation time T_60 = 0.161 V/(A α), modal density, diffraction, absorption.
Sound speed in an ideal gas
c = √(γ R T / M), where γ = c_p/c_v is the adiabatic index, R the gas constant, T temperature, and M molar mass. Derives from treating…
Phononic crystals & acoustic metamaterials
Periodic elastic media with bandgaps; negative mass density / bulk modulus; cloaking + super-resolution imaging. Analog of photonic…
Psychoacoustics & masking
Equal-loudness contours (Fletcher-Munson), critical bands, simultaneous/temporal masking. Foundation of MP3/AAC perceptual audio coding.
Underwater acoustics & SONAR
SOFAR channel guides low-f waves across ocean basins. Active/passive SONAR; ambient noise (wind, shipping, biological). Target strength +…
Linear acoustic wave equation
∂²p/∂t² = c²∇²p for small-amplitude pressure fluctuations; c = √(γP/ρ) in gas, √(K/ρ) in liquid.
Acoustic impedance & matching
Z = ρc; transmission coefficient 4Z₁Z₂/(Z₁+Z₂)²; tympanic membrane matches air to cochlea.
Helmholtz resonator
Cavity-neck system with f₀ = (c/2π)√(A/(VL')); muffler, wine-bottle whistle; harmonic filters.
Sabine reverberation formula
T₆₀ = 0.161 V/(Aα̅) in m, s; balances absorption against volume; basis of concert-hall design.
Rayleigh integral for piston radiation
p(r) = (jωρ/2π) ∫∫_S v_n e^(-jkR)/R dS; diffraction pattern of piston source; basis of beam forming.
Acoustic Doppler & sonar
f' = f(c+v_obs)/(c-v_src) classical limit; Doppler radar/sonar uses reflected-signal shift to measure velocity; blood-flow Doppler.
Nonlinear acoustics & Burgers' equation
∂p/∂x + (β/c³)p ∂p/∂τ = (δ/c³)∂²p/∂τ²; shock formation, sonic boom; limit of wave equation at large amplitude.
Biot theory of porous media
Two-phase (solid, fluid) Lamé-like equations; predicts slow and fast compressional waves; basis of seismic exploration.
Sonar, SONAR equation
Active: SL - 2TL + TS - NL + DI > DT; passive omits TS, halves TL; applies in ocean channels, SOFAR.
Medical ultrasound & B-mode imaging
1-20 MHz pulses, echo times map tissue; Doppler blood flow; elastography; safety limit MI<1.9.
Cavitation & Rayleigh–Plesset
R R̈ + (3/2)Ṙ² = (P_B-P_∞)/ρ - 4μṘ/R - 2σ/(ρR); governs bubble dynamics, sonoluminescence.
Sonoluminescence
Collapsing bubble emits ps-duration UV flash; temperature ~10⁴-10⁵ K; plasma formation; still debated fusion speculation.
Phononic crystals & metamaterials
Periodic elastic structures with bandgaps analogous to photonic; negative mass/bulk modulus at resonance; cloaking.
SOFAR sound channel
Minimum-speed depth ~1 km traps low-frequency sound; whales and submarines exploit for long-range propagation.
Lighthill aeroacoustic analogy
Sound source T_ij = ρv_iv_j + p'δ_ij - τ_ij from turbulent flow; jet noise P ∝ U⁸ (Lighthill's 8th-power law).
Elastic waves (P and S)
Longitudinal P-wave v_P=√((λ+2μ)/ρ); shear S-wave v_S=√(μ/ρ); Rayleigh surface waves on free surface.
Surface acoustic waves (SAW) devices
Rayleigh SAW on piezoelectric substrate (LiNbO₃); interdigital transducers; GHz filters in cellphones, chemical sensors.
Psychoacoustics & Fletcher–Munson
Equal-loudness contours showing ear's frequency sensitivity peaked near 3-4 kHz; basis of dB(A) weighting, MP3 coding.