Unit 1: BASIC ACOUSTIC PRINCIPLES

I. Foundations and Orientation

Acoustics studies the generation, propagation, interaction, measurement, and effects of mechanical disturbances in material media. Classical aeroacoustics usually treats air as a continuous, compressible fluid and represents sound as small perturbations superimposed on an equilibrium or moving mean flow.

A. Basic acoustic principles

Basic acoustic analysis connects pressure, density, and particle-motion disturbances through conservation laws and a thermodynamic closure relation.

• Continuum assumption: Pressure (p), density (\rho), and velocity (\mathbf{u}) vary continuously over distances much larger than the molecular mean free path.
• Small-disturbance assumption: Each quantity is decomposed into mean and acoustic parts:
  
  TEXT

    p = p₀ + p′,    ρ = ρ₀ + ρ′,    u = U₀ + u′

  
  Here (p_0,\rho_0,\mathbf U_0) are mean values, while (p',\rho',\mathbf u') are perturbations.
• Linear acoustics: When (|p'|\ll p_0), (|\rho'|\ll\rho_0), and acoustic particle speed is much smaller than sound speed, products of perturbations can be neglected.
• Compressibility: Sound exists because pressure changes produce density changes; perfectly incompressible media do not support ordinary longitudinal acoustic waves.
• Adiabatic propagation: Rapid acoustic compression normally permits negligible heat transfer, giving the sound speed
  
  TEXT

    c² = (∂p/∂ρ)ₛ

  
  where (c) is sound speed and the derivative is evaluated at constant entropy (s).
• Superposition: In a linear medium, individual acoustic solutions add without changing one another.

II. Nature of Sound

A. Sound and sound waves

Sound is a mechanical disturbance that transports energy through a medium by coupled pressure, density, and particle-velocity fluctuations.

• Longitudinal propagation: In gases, particles oscillate mainly parallel to the direction of wave travel, creating alternating compressions and rarefactions.
• Harmonic plane wave: A sinusoidal pressure disturbance travelling in the (+x)-direction is
  
  TEXT

    p′(x,t) = p̂ cos(ωt − kx + φ)

  
  where (\hat p) is peak pressure, (\omega=2\pi f) is angular frequency, (f) is frequency, (k=2\pi/\lambda) is wavenumber, (\lambda) is wavelength, and (\phi) is phase.
• Propagation speed: The phase speed satisfies
  
  TEXT

    c = fλ = ω/k

  
  For dry air near (20^\circ\text{C}), (c) is approximately (343\ \text{m s}^{-1}).
• Wave types: Plane waves approximate distant sources; spherical waves radiate from compact sources and decrease in pressure amplitude approximately as (1/r), where (r) is source distance.
• Audible range: Human hearing conventionally spans about (20\ \text{Hz}) to (20\ \text{kHz}), although sensitivity varies with frequency and age.

III. Wave Bending

A. Diffraction

Diffraction is the bending and spreading of sound around obstacles or through openings when geometrical-ray propagation is insufficient.

• Controlling ratio: Diffraction is strong when wavelength (\lambda) is comparable to or larger than obstacle dimension (D); it is weak when (\lambda\ll D).
• Frequency dependence: Low-frequency sound has a long wavelength and therefore bends around barriers more effectively than high-frequency sound.
• Edge diffraction: A barrier edge behaves like a secondary radiator, allowing acoustic energy to enter the geometrical shadow region.
• Aperture behavior: An opening much smaller than (\lambda) radiates broadly, while an opening many wavelengths wide produces a narrow transmitted beam.
• Aeroacoustic significance: Diffraction alters aircraft-engine shielding, airframe-noise directivity, and measured sound levels when structures lie between source and receiver.

IV. Boundary Interaction

A. Reflection

Reflection occurs when a sound wave encounters a boundary across which acoustic impedance changes.

• Acoustic impedance: For a plane progressive wave, characteristic impedance is (Z=\rho c), measured in (\text{Pa·s m}^{-1}).
• Normal-incidence coefficient: At a boundary between impedances (Z_1) and (Z_2),
  
  TEXT

    R = p̂ᵣ/p̂ᵢ = (Z₂ − Z₁)/(Z₂ + Z₁)

  
  where (\hat p_i) and (\hat p_r) are incident and reflected pressure amplitudes.
• Energy reflection: For lossless media at normal incidence, the reflected intensity fraction is (|R|^2).
• Rigid boundary: Particle velocity normal to an ideal rigid wall is zero; pressure reflection occurs without a pressure-phase reversal.
• Interference: Incident and reflected waves can form standing waves with pressure nodes and antinodes separated by (\lambda/4).

V. Spatial Acoustic Conditions

A. Sound field

A sound field is the spatial and temporal distribution of acoustic pressure, particle velocity, intensity, and related quantities.

• Free field: Sound propagates without significant reflections; an ideal point source exhibits inverse-square intensity decay, (I\propto1/r^2).
• Diffuse field: Reflections produce approximately equal acoustic energy density in all directions, as approximated in a reverberation chamber.
• Near field: Close to a source, reactive energy storage can be important, and pressure and particle velocity need not be in phase.
• Far field: At sufficiently large distance, waves are predominantly radiating, pressure and velocity are nearly in phase, and source directivity becomes stable.
• Mean-flow field: In aeroacoustics, convection and refraction by nonuniform flow can alter propagation speed, wavelength, and radiation direction.

VI. Acoustic Quantities

A. Acoustic terminology and definitions

Acoustic terminology provides consistent distinctions among instantaneous quantities, amplitudes, spectra, and logarithmic levels.

• Sound pressure: (p'=p-p_0), measured in pascals (Pa), is the deviation from mean static pressure.
• Particle velocity: (\mathbf u'), in (\text{m s}^{-1}), is the oscillatory fluid velocity and is distinct from sound-wave speed (c).
• Frequency and period: (f) is cycles per second in hertz, while period (T=1/f).
• RMS value: For a sinusoid, (p_{\mathrm{rms}}=\hat p/\sqrt2); RMS values represent equivalent energetic magnitude.
• Sound-pressure level:
  
  TEXT

    Lp = 20 log₁₀(pᵣₘₛ/pref) dB

  
  where (p_{\mathrm{ref}}=20\ \mu\text{Pa}) in air.
• Spectrum: A spectrum distributes pressure or energy by frequency; octave and one-third-octave bands provide standardized bandwidths.
• Directivity: Source directivity describes variation of radiated sound with angle and is often expressed relative to a chosen axis.

VII. Wave Kinematics and Energy

A. Relationship between wavelengths, particle velocities, acceleration, energy density, and acoustic intensity

For a progressive plane wave, wavelength sets spatial scale, while particle motion and pressure jointly determine stored and transported acoustic energy.

• Wavelength relation:
  
  TEXT

    λ = c/f

  
  Thus a (1\ \text{kHz}) tone in air at (c=343\ \text{m s}^{-1}) has (\lambda=0.343\ \text{m}).
• Pressure–velocity relation:
  
  TEXT

    p̂ = ρ₀c û

  
  where (\hat u) is peak particle velocity parallel to propagation.
• Particle acceleration: For (u'=\hat u\cos(\omega t-kx)), acceleration is (a'=\partial u'/\partial t), with peak value (\hat a=\omega\hat u).
• Instantaneous energy density:
  
  TEXT

    e = p′²/(2ρ₀c²) + ρ₀u′²/2

  
  The terms are compressional potential energy and particle kinetic energy per unit volume, in (\text{J m}^{-3}).
• Acoustic intensity:
  
  TEXT

    I = ⟨p′u′⟩ = pᵣₘₛ²/(ρ₀c) = ρ₀c uᵣₘₛ²

  
  Here (\langle\cdot\rangle) denotes time averaging, and (I) is measured in (\text{W m}^{-2}).
• Energy transport: For a plane progressive wave, mean intensity equals mean total energy density multiplied by (c).

VIII. Acoustic Metrology

A. Reference standards and measurement

Acoustic measurement compares calibrated electrical or digital signals with internationally defined reference quantities and instrument requirements.

• Pressure reference: Airborne sound-pressure level uses (20\ \mu\text{Pa}); underwater acoustics commonly uses (1\ \mu\text{Pa}), so levels require their reference to be stated.
• Intensity and power references: Common references are (10^{-12}\ \text{W m}^{-2}) for intensity and (10^{-12}\ \text{W}) for sound power.
• Instrumentation: A measurement chain typically includes a microphone, preamplifier, signal conditioner, analogue-to-digital converter, and spectral analyser.
• Calibration: An acoustic calibrator applies a known pressure level and frequency to verify microphone sensitivity before and after measurements.
• Standards: IEC 61672 specifies sound-level-meter performance; ISO 3744 addresses sound-power determination using sound pressure over a reflecting plane.
• Frequency weighting: A-weighting approximates human sensitivity at moderate levels; unweighted or Z-weighted data preserve broad-band physical magnitude.
• Uncertainty controls: Microphone position, wind noise, reflections, background noise, averaging time, and calibration drift must be documented.

IX. Fluid-Dynamic Basis

A. Dynamics of acoustic principles

Acoustic dynamics arise by applying mass and momentum conservation to a compressible fluid and then linearizing about a mean state.

• Governing variables: Pressure (p), density (\rho), velocity (\mathbf u), and entropy (s) describe the local fluid state.
• Material derivative:
  
  TEXT

    D/Dt = ∂/∂t + u·∇

  
  It measures change following a moving fluid particle.
• Linearization: Substituting (p=p_0+p'), (\rho=\rho_0+\rho'), and (\mathbf u=\mathbf u') converts nonlinear conservation laws into coupled linear acoustic equations for a stationary uniform medium.
• Nonlinearity: At large amplitudes, wave speed varies locally with state, causing waveform steepening and potentially shock formation.
• Mean-flow effects: A nonzero (\mathbf U_0) introduces convection; acoustic disturbances are transported by both fluid motion and propagation relative to the fluid.

X. Conservation of Mass

A. Continuity Equation

The continuity equation states that fluid mass cannot be created or destroyed within a material control volume.

• Exact differential form:
  
  TEXT

    ∂ρ/∂t + ∇·(ρu) = 0

  
  where (\nabla\cdot) denotes divergence.
• Material form:
  
  TEXT

    Dρ/Dt + ρ∇·u = 0

  
  Density increases when a fluid element undergoes negative volumetric expansion.
• Linear acoustic form: For uniform, stationary (\rho_0),
  
  TEXT

    ∂ρ′/∂t + ρ₀∇·u′ = 0

• Physical meaning: Converging particle velocity produces compression, while diverging velocity produces rarefaction.
• Applicability: The exact equation remains valid for nonlinear acoustics; only the reduced form depends on the small-disturbance assumptions.

XI. Conservation of Momentum

A. Euler’s Equation

Euler’s equation expresses momentum conservation for an inviscid fluid subjected to pressure forces and, when present, body forces.

• Exact inviscid form:
  
  TEXT

    ρ Du/Dt = −∇p + ρb

  
  where (\mathbf b) is body force per unit mass.
• Linear acoustic form: With no body force and a stationary uniform medium,
  
  TEXT

    ρ₀ ∂u′/∂t = −∇p′

• Pressure-gradient role: Particle acceleration points from high acoustic pressure toward low acoustic pressure.
• Harmonic implication: For a plane progressive wave, combining Euler’s equation with (k=\omega/c) gives (\hat p=\rho_0c\hat u).
• Limitation: Viscous stresses are omitted; the Navier–Stokes equation is required when boundary layers, attenuation, or small passages make viscosity significant.

XII. Thermodynamic Closure

A. Poisson’s Equation

Poisson’s equation for an ideal gas supplies the adiabatic pressure–density relation needed to close the acoustic conservation equations.

• Adiabatic relation:
  
  TEXT

    p/ρᵞ = constant

  
  where (\gamma=c_p/c_v) is the ratio of specific heats.
• Equivalent state relation:
  
  TEXT

    p = Kρᵞ

  
  Here (K) remains constant during an isentropic disturbance.
• Linearized relation:
  
  TEXT

    p′ = c²ρ′

  
  where (c^2=(\partial p/\partial\rho)_s).
• Ideal-gas sound speed:
  
  TEXT

    c = √(γp₀/ρ₀) = √(γRT₀)

  
  Here (R) is specific gas constant and (T_0) is absolute mean temperature.
• Interpretation: Sound speed rises approximately with (\sqrt{T_0}); mean pressure alone does not determine (c) when temperature is fixed.

XIII. Propagation Equation

A. Wave Equation

The wave equation describes how acoustic disturbances propagate through a uniform, stationary, lossless medium.

• Pressure form:
  
  TEXT

    ∇²p′ − (1/c²) ∂²p′/∂t² = 0

  
  where (\nabla^2) is the Laplacian operator.
• Derivation: Taking the divergence of linearized Euler’s equation, differentiating continuity in time, and using (p'=c^2\rho') eliminates (\mathbf u') and (\rho').
• Velocity-potential form: If (\mathbf u'=\nabla\phi), then
  
  TEXT

    ∇²φ − (1/c²) ∂²φ/∂t² = 0

  
  where (\phi) is acoustic velocity potential.
• One-dimensional solution:
  
  TEXT

    p′(x,t) = F(x − ct) + G(x + ct)

  
  Functions (F) and (G) represent waves travelling in the (+x) and (-x) directions.
• Source inclusion: Real aeroacoustic problems use an inhomogeneous wave equation whose source terms represent unsteady forces, mass injection, turbulence, or fluctuating stresses.
• Model limits: Uniform-medium form excludes mean-flow gradients, strong nonlinearity, viscosity, thermal conduction, and atmospheric refraction unless additional terms are introduced.