Chapter - 11

Key Words: Attenuation, Sound absorption, Acoustic propagation, Medium properties

Attenuation of Sound Waves

🎯 Learning Objectives

By the end of this chapter, you should be able to:

  • Explain the physical mechanisms responsible for sound attenuation in fluids and solids.
  • Distinguish between geometric spreading, absorption, and scattering contributions to total attenuation.
  • Apply formulas for frequency-dependent absorption and viscous/thermal losses.
  • Compute attenuation coefficients for common materials and atmospheric conditions.
  • Analyze propagation in layered media, porous materials, and acoustic filters.

Introduction

As sound propagates through a medium, its amplitude decreases due to a combination of geometric spreading, viscous and thermal losses, and scattering by particles or inhomogeneities. This process is collectively called attenuation.

Attenuation is important in:

  • Architectural acoustics (wall and ceiling materials)
  • Underwater acoustics
  • Noise control engineering
  • Atmospheric sound propagation

The *pressure amplitude of a plane wave traveling along a lossy medium can be described as:

p(x)=p0eβˆ’Ξ±xej(Ο‰tβˆ’kx)p(x) = p_0 e^{-\alpha x} e^{j(\omega t - kx)}

Where:

  • p0p_0 is the initial pressure amplitude
  • Ξ±\alpha is the attenuation coefficient [Np/mNp/m]
  • kk is the wavenumber
  • xx is the propagation distance

Physical Mechanisms of Attenuation

Geometric Spreading

Even in an ideal lossless medium, amplitude decreases due to energy spreading over a larger area:

  • Spherical wave:
    p(r)∼1rp(r) \sim \frac{1}{r}
  • Cylindrical wave:
    p(r)∼1rp(r) \sim \frac{1}{\sqrt{r}}

Viscous and Thermal Losses

In fluids, sound loses energy due to viscosity and heat conduction:

Ξ±viscous+Ξ±thermal=Ο‰22ρ0c3(4Ξ·3+ΞΆ+(Ξ³βˆ’1)ΞΊ)\alpha_{viscous} + \alpha_{thermal} = \frac{\omega^2}{2 \rho_0 c^3} \left( \frac{4 \eta}{3} + \zeta + (\gamma - 1) \kappa \right)

Where:

  • Ξ·\eta = shear viscosity
  • ΞΆ\zeta = bulk viscosity
  • ΞΊ\kappa = thermal conductivity
  • Ξ³\gamma = ratio of specific heats

Relaxation Processes

Atmospheric gases can absorb energy via molecular relaxation. Oxygen and nitrogen molecules store energy temporarily in vibrational or rotational modes:

Ξ±relax=Ο‰22c3ρ0βˆ‘iΟ„i1+(ωτi)2Ξ”Ei\alpha_{relax} = \frac{\omega^2}{2 c^3 \rho_0} \sum_i \frac{\tau_i}{1 + (\omega \tau_i)^2} \Delta E_i

Where Ο„i\tau_i is the relaxation time of the i-th process.

Scattering

Inhomogeneities like bubbles, dust, or rough boundaries redirect energy out of the original wave path. This can be described via scattering cross-section, Οƒs\sigma_s:

αscattering=nσs\alpha_{scattering} = n \sigma_s

Where nn is the number density of scatterers.

Frequency Dependence

Attenuation generally increases with frequency. High-frequency components are more strongly absorbed:

  • Air: α∼f2\alpha \sim f^2 (dominant molecular absorption at high frequencies)
  • Water: depends on salinity, temperature, and pressure
  • Porous materials: α∼f\alpha \sim \sqrt{f} approximately for fibrous absorbers

Example: At 1kHz1 kHz, air at 20Β°C20Β°C and 50%50\% humidity has Ξ±β‰ˆ0.01 dB/m\alpha \approx 0.01 \, \text{dB/m}, while at 10kHz10 kHz, Ξ±β‰ˆ1.0 dB/m\alpha \approx 1.0 \, \text{dB/m}.

Practical Formulas

Amplitude Decay (dB)

L(x)=20log⁑10p0p(x)=20log⁑10e αxβ‰ˆ8.686 αxL(x) = 20 \log_{10} \frac{p_0}{p(x)} = 20 \log_{10} e \, \alpha x \approx 8.686 \, \alpha x

Attenuation in Air (empirical)

For air, absorption depends on frequency (f, in kHz) and humidity:

Ξ±airβ‰ˆf2(f2+fr2)Γ—C dB/m\alpha_{air} \approx \frac{f^2}{(f^2 + f_r^2)} \times C \, \text{dB/m}

Where frf_r is relaxation frequency (~1–10 kHz depending on humidity), and C is a constant.

Layered Media (acoustic impedance mismatches)

When sound propagates through layers with different impedances, part of the wave is reflected, reducing transmitted amplitude:

ptransmitted=2Z2Z1+Z2pincidentp_{transmitted} = \frac{2 Z_2}{Z_1 + Z_2} p_{incident}

Examples:

  • Fibrous absorber: Ξ±β‰ˆ0.2βˆ’1.0\alpha \approx 0.2-1.0 for 500–4000Hz500–4000 Hz
  • Water: α∼0.002βˆ’0.05 dB/m\alpha \sim 0.002-0.05 \, \text{dB/m} for typical seawater at 1–10kHz1–10 kHz
  • Atmospheric absorption: critical for outdoor propagation, especially above 2kHz2 kHz

πŸ“ Key Takeaways

  • Attenuation reduces sound amplitude due to spreading, absorption, and scattering.
  • Frequency-dependent effects are important: high frequencies are absorbed faster.
  • Physical mechanisms include viscous/thermal losses, molecular relaxation, and scattering.
  • Layered media and impedance mismatches can increase effective attenuation.
  • Engineers must account for attenuation in room acoustics, outdoor sound propagation, and underwater acoustics.

🧠 Quick Quiz

1) Which factor primarily increases sound attenuation at high frequencies in air?

2) The amplitude of a plane wave in a lossy medium decays as:

3) Which of the following increases attenuation in a medium?

4) In layered media, how is transmitted pressure amplitude affected at an interface?

5) Which physical mechanism contributes to attenuation in solids and fluids?