Unit 2: ACOUSTIC WAVE TRANSMISSION

I. Orientation — Foundations of Acoustic Transmission

Acoustic transmission is the propagation of small pressure disturbances through a compressible medium; in aeroacoustics, these disturbances often interact with atmospheric gradients, boundaries, and fluid motion.

A. Governing Framework

The linear acoustic model follows from conservation of mass, momentum, and energy applied to small perturbations about a mean state.

• Acoustic variables: Total pressure, density, and velocity are decomposed into mean and fluctuating parts:
  

  TEXT

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

  
  Here, (p_0,\rho_0,\mathbf U) are mean pressure, density, and flow velocity; (p',\rho',\mathbf u') are acoustic perturbations.

• Linear-acoustic assumptions:

  • Perturbations satisfy (|p'|\ll p_0), (|\rho'|\ll\rho_0), and (|\mathbf u'|\ll c_0).
  • The undisturbed medium is locally homogeneous and thermodynamic changes are approximately isentropic.
  • Viscous and thermal losses are initially neglected.

• Speed of sound:
  

  TEXT

    c₀² = (∂p/∂ρ)ₛ = γp₀/ρ₀ = γRT

  
  Here, (c_0) is sound speed, (s) entropy, (\gamma) the specific-heat ratio, (R) the specific gas constant, and (T) absolute temperature.

• Wave equation in a stationary uniform medium:
  

  TEXT

    ∇²p′ − (1/c₀²)(∂²p′/∂t²) = 0

  
  The operator (\nabla^2) is the Laplacian and (t) is time.

• Harmonic convention: For angular frequency (\omega=2\pi f), a field may be written (p'=\Re{\hat p e^{-i\omega t}}), with wavenumber (k=\omega/c_0=2\pi/\lambda).

II. Atmospheric Propagation — Refraction, Spreading, and Loss

A. Propagation of sound in the atmosphere

Atmospheric sound propagation differs from ideal free-field propagation because temperature, wind, humidity, turbulence, and the ground modify amplitude and direction.

• Geometrical spreading: A point source radiating spherically has pressure amplitude proportional to (1/r) and intensity proportional to (1/r^2):
  

  TEXT

    I(r) = W/(4πr²)

  
  Here, (I) is intensity, (W) acoustic power, and (r) source distance. Doubling (r) ideally reduces sound-pressure level by (6\ \text{dB}).

• Atmospheric absorption: Molecular relaxation and classical viscous-thermal effects produce frequency-dependent attenuation:
  

  TEXT

    Lp(r₂) = Lp(r₁) − 20log₁₀(r₂/r₁) − α(r₂ − r₁)

  
  Here, (L_p) is sound-pressure level in decibels and (\alpha) is atmospheric attenuation in (\text{dB m}^{-1}). High frequencies generally attenuate more rapidly.

• Temperature refraction: Since (c_0\approx\sqrt{\gamma RT}), sound speed rises with temperature. Rays bend toward regions of lower effective sound speed; a normal daytime temperature decrease with altitude can bend rays upward and create a shadow zone.

• Wind refraction: For propagation direction (\mathbf n), the effective ray speed is approximately
  

  TEXT

    c_eff = c₀ + U·n

  
  Downwind rays commonly bend toward the ground, whereas upwind rays bend upward.

• Ground interaction: Direct and ground-reflected waves interfere. Their phase difference depends on path length, frequency, incidence angle, and complex ground impedance; porous soil generally absorbs more sound than rigid concrete.

• Turbulence and weather: Random temperature and velocity fluctuations scatter sound, blur interference minima, and cause short-term level variations.

B. Applications and Limitations

Atmospheric models support aircraft-noise prediction but become approximate when weather and terrain vary strongly.

• Applications: Airport noise contours, sonic-boom propagation, community-noise assessment, and long-range detection use spreading, absorption, and meteorological corrections.
• Limitations: A single constant (\alpha) or straight-ray model cannot accurately represent layered winds, turbulence, irregular terrain, or rapidly changing humidity.

III. Moving-Medium Sources — Convection and Doppler Effects

A. Sound sources in moving media

Relative motion between a source, observer, and fluid changes wavefront geometry, observed frequency, and radiation strength.

• Uniform mean flow: For constant flow (\mathbf U), acoustic pressure satisfies the convected wave equation:
  

  TEXT

    (∂/∂t + U·∇)²p′ − c₀²∇²p′ = 0

  
  The material operator (\partial/\partial t+\mathbf U\cdot\nabla) accounts for convection by the mean flow.

• Moving-source Doppler shift: For a stationary observer and a source moving at velocity (\mathbf V_s),
  

  TEXT

    f_obs = f_s/[1 − (V_s·n)/c₀]

  
  Here, (fs) is emitted frequency, (f{\text{obs}}) observed frequency, and (\mathbf n) points from the source toward the observer at emission. Approach increases frequency; recession decreases it.

• Mach number: Source speed is classified by (M_s=|\mathbf V_s|/c_0). Subsonic sources have (M_s<1), while supersonic sources have (M_s>1).

• Mach cone: A supersonic source produces an envelope of wavefronts with half-angle
  

  TEXT

    μ = sin⁻¹(1/Mₛ)

  
  Here, (\mu) is the Mach angle. The associated pressure disturbance can be perceived as a sonic boom.

• Retarded time: Observer pressure depends on the source at an earlier emission time (t_r):
  

  TEXT

    t − tᵣ = |x − y(tᵣ)|/c₀

  
  Here, (\mathbf x) is observer position and (\mathbf y(t_r)) is source position at emission.

B. Applications and Limitations

Moving-source theory connects source kinematics to flyover noise and high-speed aerodynamic radiation.

• Applications: Aircraft flyovers, rotating blades, propellers, jet structures, and railway noise require Doppler and retarded-time corrections.
• Limitations: Simple formulas assume uniform flow, constant source velocity, compact sources, and negligible atmospheric gradients.

IV. Integral Representation — Green-Function Method

A. Generalized Green’s formula

Generalized Green’s formula represents an acoustic field through volume sources, boundary data, and a Green function satisfying the governing operator.

• Green function: For the Helmholtz operator, (G(\mathbf x,\mathbf y)) is defined by
  

  TEXT

    (∇²_y + k²)G(x,y) = −δ(x − y)

  
  Here, (\mathbf x) is the observation point, (\mathbf y) the source point, (k) the wavenumber, and (\delta) the Dirac delta.

• Free-space solution:
  

  TEXT

    G(x,y) = exp(ikR)/(4πR),    R = |x − y|

  
  This is the outgoing spherical-wave response to a unit point source.

• Generalized representation: If ((\nabla^2+k^2)\hat p=-\hat q) in volume (V), then
  

  TEXT

    p̂(x) = ∫V Gq̂ dV
            + ∮S [G(∂p̂/∂n) − p̂(∂G/∂n)] dS

  
  Here, (\hat p) is complex pressure amplitude, (\hat q) a volume-source distribution, (S) the boundary, and (\partial/\partial n) the outward normal derivative.

• Physical interpretation:

  1. Volume term: (\int_VG\hat q\,dV) propagates distributed source effects.
  2. Surface term: The boundary integral acts as equivalent monopole and dipole source distributions.

• Boundary conditions: A rigid wall imposes (\partial\hat p/\partial n=0); a pressure-release boundary imposes (\hat p=0). A suitably constructed Green function can satisfy these conditions automatically.

B. Significance and Limitations

The formula converts a differential wave problem into an integral radiation problem.

• Significance: It underlies boundary-element methods, Kirchhoff surface formulations, scattering calculations, and acoustic analogies.
• Limitations: Numerical evaluation becomes expensive for large surfaces, and singular kernels require careful integration near (\mathbf x=\mathbf y).

V. Aerodynamic Sound Generation — Acoustic Analogy

A. Lighthill equation

Lighthill’s equation rearranges the exact compressible-flow equations into a wave equation driven by fluctuating flow stresses.

• Formal equation:
  

  TEXT

    ∂²ρ/∂t² − c₀²∇²ρ = ∂²Tᵢⱼ/(∂xᵢ∂xⱼ)

  
  Here, (\rho) is density, (c_0) a chosen ambient sound speed, (x_i) Cartesian coordinates, and repeated indices imply summation.

• Lighthill stress tensor:
  

  TEXT

    Tᵢⱼ = ρuᵢuⱼ + (p − c₀²ρ)δᵢⱼ − τᵢⱼ

  
  Here, (ui) are velocity components, (p) pressure, (\delta{ij}) the Kronecker delta, and (\tau_{ij}) viscous stress.

• Source character: The double divergence of (T_{ij}) has quadrupole form. In high-Reynolds-number, nearly isentropic free turbulence, the Reynolds-stress term (\rho u_i u_j) is often dominant.

• Integral interpretation: Using the retarded free-space Green function gives far-field density fluctuations generated by earlier source events:
  

  TEXT

    ρ′(x,t) = (1/4πc₀²) ∂²/∂xᵢ∂xⱼ
              ∫V [Tᵢⱼ(y,t − R/c₀)/R] dV_y

  
  Here, (R=|\mathbf x-\mathbf y|), and brackets indicate evaluation at retarded time.

• Velocity scaling: For compact, low-Mach free turbulence, quadrupole acoustic power follows the approximate eighth-power law (W\propto U^8), explaining the strong noise increase with jet velocity (U).

B. Applications and Limitations

Lighthill’s analogy provides the foundational framework for predicting noise generated by unsteady flow.

• Applications: It is central to jet-noise theory and extends to solid-boundary formulations that identify monopole, dipole, and quadrupole contributions.
• Limitations: The equation is exact as a rearrangement, but prediction still requires accurate source data; separating “source” from “propagation” becomes difficult in nonuniform mean flow.

VI. Resonant Acoustic Fields — Nodes and Antinodes

A. Standing waves and standing wave apparatus

Standing waves result from interference between waves of equal frequency traveling in opposite directions, producing fixed nodes and antinodes.

• Wave superposition:
  

  TEXT

    p′(x,t) = 2P cos(kx) cos(ωt)

  
  Here, (P) is the amplitude of each traveling pressure wave, (x) is axial position, (k) is wavenumber, and (\omega) is angular frequency.

• Pressure pattern: Pressure antinodes occur where (|\cos(kx)|=1); pressure nodes occur where (\cos(kx)=0). Adjacent nodes are separated by (\lambda/2).

• Velocity relation: Pressure antinodes coincide with particle-velocity nodes, while pressure nodes coincide with velocity antinodes.

• Tube resonances:

  1. Open–open or closed–closed tube:
     
     TEXT

          fₙ = nc₀/(2L),    n = 1,2,3,...

  2. Open–closed tube:
     
     TEXT

          fₙ = (2n−1)c₀/(4L),    n = 1,2,3,...

     
     Here, (L) is effective tube length and (n) is mode number.

• Standing-wave apparatus: A Kundt’s tube or impedance tube uses a loudspeaker, uniform tube, termination, and movable microphone or probe. Measuring node spacing gives (\lambda=2\Delta x), followed by (c_0=f\lambda).

• Standing-wave ratio:
  

  TEXT

    SWR = |p|max/|p|min = (1 + |R|)/(1 − |R|)

  
  Here, (R) is the pressure reflection coefficient; (R=0) indicates no standing-wave modulation and (|R|=1) perfect reflection.

B. Applications and Limitations

Standing-wave measurements reveal resonance, sound speed, absorption, and boundary impedance.

• Applications: Impedance tubes determine material absorption and reflection coefficients under controlled normal incidence.
• Limitations: End corrections, probe disturbance, damping, and higher transverse modes cause deviations from one-dimensional theory.

VII. Directional Radiation — Angular Concentration of Sound

A. Beam width and directivity index

Beam width measures the angular extent of a principal radiation lobe, while directivity index measures concentration relative to an omnidirectional source.

• Radiation pattern: The normalized directivity function is
  

  TEXT

    D(θ,φ) = |p(r,θ,φ)|/|p|max

  
  Here, (\theta,\phi) are angular coordinates and (r) is sufficiently large for far-field behavior.

• Beam width: Half-power beam width is the angle between directions where intensity falls to one-half its peak, equivalent to a pressure decrease of (3\ \text{dB}).

• Directivity factor:
  

  TEXT

    Q = 4πI(θ₀,φ₀)/∫₄π I(θ,φ)dΩ

  
  Here, (I(\theta_0,\phi_0)) is intensity in the reference direction and (d\Omega) is solid angle. An omnidirectional source has (Q=1).

• Directivity index:
  

  TEXT

    DI = 10log₁₀Q  dB

  
  Thus, (Q=2) gives (DI\approx3\ \text{dB}), while (Q=10) gives (DI=10\ \text{dB}).

• Circular piston pattern:
  

  TEXT

    D(θ) = 2J₁(ka sinθ)/(ka sinθ)

  
  Here, (J_1) is the first-order Bessel function and (a) is piston radius. Increasing (ka), or aperture size relative to wavelength, narrows the main beam.

• Interpretation: Narrow beam width generally corresponds to larger (Q) and (DI), although sidelobes mean beam width alone does not determine total radiated directivity.

B. Applications and Limitations

Directional measures characterize aircraft sources, loudspeakers, acoustic arrays, and ultrasonic transducers.

• Applications: Polar plots and directivity indices support noise mapping, microphone placement, array design, and source identification.
• Limitations: Directivity varies with frequency, operating condition, installation geometry, and near-field distance; far-field values cannot automatically describe measurements close to the source.