Sound propagation in different media is crucial for understanding how acoustic waves behave in various environments. This topic covers how sound travels through air, solids, liquids, and porous materials, exploring factors that affect speed and attenuation.

The chapter delves into key concepts like wave types, acoustic impedance, and transmission loss. It also examines phenomena such as diffraction, scattering, refraction, and diffusion, which are essential for predicting sound fields in complex architectural spaces.

Sound propagation in air

• Understanding how sound waves travel through air is crucial for architectural acoustics to optimize room acoustics and minimize noise transmission
• Speed of sound in air and factors affecting it determine how sound behaves in various environments and conditions
• Attenuation and air absorption coefficients are key to predicting sound levels at different distances from the source

Speed of sound in air

• Approximately 343 meters per second (m/s) at 20°C (68°F) and sea level
• Increases with temperature at a rate of about 0.6 m/s per degree Celsius
• Not significantly affected by air pressure or humidity under normal conditions
• Can be calculated using the formula: $c = \sqrt{\frac{\gamma R T}{M}}$, where $\gamma$ is the adiabatic index, $R$ is the universal gas constant, $T$ is the absolute temperature, and $M$ is the molar mass of air

Factors affecting sound speed

• Temperature is the primary factor affecting sound speed in air
  • Higher temperatures lead to faster sound propagation
  • A temperature gradient can cause refraction of sound waves
• Humidity has a minor effect on sound speed
  • Increasing humidity slightly increases sound speed
• Composition of the gas mixture (e.g., air with higher CO2 content) can affect sound speed

Attenuation of sound in air

• Sound intensity decreases with distance from the source due to geometric spreading and air absorption
• Geometric spreading is the primary cause of attenuation in free field conditions
  • Sound intensity decreases by 6 dB per doubling of distance for a point source (inverse square law)
  • Sound intensity decreases by 3 dB per doubling of distance for a line source
• Air absorption is caused by viscous losses, thermal conduction, and molecular relaxation
  • More significant at higher frequencies and longer distances
  • Dependent on temperature, humidity, and pressure

Air absorption coefficients

• Measure of the sound energy absorbed by air per unit distance
• Expressed in decibels per meter (dB/m) or per kilometer (dB/km)
• Vary with frequency, temperature, humidity, and pressure
• Can be calculated using standardized equations (e.g., ISO 9613-1)
• Higher frequencies experience greater absorption
  • Important consideration for long-distance sound propagation and high-frequency noise control

Sound propagation in solids

• Solid materials are often used in architectural acoustics for sound insulation, structural components, and acoustic treatments
• Understanding sound propagation in solids is essential for designing effective noise control and enhancing sound quality in buildings
• Longitudinal and transverse waves, speed of sound, and attenuation in solids are key concepts for predicting sound transmission and vibration behavior

Longitudinal vs transverse waves

• Longitudinal waves (also called compression waves or P-waves) propagate parallel to the direction of wave motion
  • Particles oscillate back and forth in the same direction as the wave
  • Cause compression and rarefaction of the material
• Transverse waves (also called shear waves or S-waves) propagate perpendicular to the direction of wave motion
  • Particles oscillate perpendicular to the wave direction
  • Cause shearing deformation of the material
• Solids can support both longitudinal and transverse waves, while fluids (like air) only support longitudinal waves

Speed of sound in solids

• Depends on the material properties, such as density and elastic moduli
• Longitudinal wave speed: $c_L = \sqrt{\frac{E}{\rho}}$, where $E$ is the Young's modulus and $\rho$ is the density
• Transverse wave speed: $c_T = \sqrt{\frac{G}{\rho}}$, where $G$ is the shear modulus
• Generally higher than the speed of sound in air (e.g., ~5000 m/s in steel, ~1500 m/s in concrete)

Factors affecting sound speed

• Material properties: density, Young's modulus, and shear modulus
  • Stiffer materials (higher moduli) have higher sound speeds
  • Denser materials have lower sound speeds
• Temperature: sound speed generally increases with temperature in solids
• Anisotropy: sound speed may vary with the direction of propagation in anisotropic materials (e.g., wood)

Attenuation of sound in solids

• Caused by various mechanisms, including viscous losses, thermal conduction, and scattering
• Depends on the material properties, frequency, and temperature
• Generally increases with frequency and temperature
• Can be expressed using the attenuation coefficient (dB/m) or loss factor (dimensionless)

Transmission loss through solids

• Measure of the sound insulation provided by a solid material or structure
• Expressed in decibels (dB) as a function of frequency
• Depends on the material properties, thickness, and incident angle of the sound waves
• Can be measured using standardized test methods (e.g., ISO 10140) or predicted using models (e.g., mass law, coincidence effect)
• Higher transmission loss indicates better sound insulation

Sound propagation in liquids

• Liquids, such as water, are important media for sound propagation in various applications, including underwater acoustics and hydraulic systems
• Understanding sound propagation in liquids is essential for designing effective underwater communication, sonar systems, and noise control in liquid-filled systems
• Speed of sound, factors affecting it, and attenuation in liquids are key concepts for predicting sound behavior in liquid media

Speed of sound in liquids

• Depends on the liquid properties, such as density and bulk modulus
• Can be calculated using the formula: $c = \sqrt{\frac{K}{\rho}}$, where $K$ is the bulk modulus and $\rho$ is the density
• Generally higher than the speed of sound in air but lower than in solids (e.g., ~1500 m/s in water, ~1400 m/s in oil)
• Varies with temperature, pressure, and salinity (for water)

Factors affecting sound speed

• Temperature: sound speed generally increases with temperature in liquids
  • In water, sound speed increases by ~3 m/s per degree Celsius
• Pressure: sound speed increases with pressure in liquids
  • In water, sound speed increases by ~1.6 m/s per 100 m depth increase
• Salinity (for water): sound speed increases with salinity
  • A 1 PSU (Practical Salinity Unit) increase in salinity leads to a ~1.3 m/s increase in sound speed

Attenuation of sound in liquids

• Caused by viscous losses, thermal conduction, and molecular relaxation
• Depends on the liquid properties, frequency, temperature, and pressure
• Generally increases with frequency and decreases with temperature
• Can be expressed using the attenuation coefficient (dB/m) or absorption coefficient (dimensionless)
• Lower than attenuation in air at the same frequency, allowing for longer-range propagation

Underwater acoustics applications

• Sonar (Sound Navigation and Ranging): uses sound waves to detect and locate objects underwater
  • Active sonar: emits sound pulses and listens for echoes
  • Passive sonar: listens for sound emitted by targets
• Underwater communication: uses acoustic waves to transmit data between underwater devices (e.g., modems, sensors)
• Seismic exploration: uses sound waves to map subsurface geological structures
• Marine mammal monitoring: uses passive acoustic monitoring to study and track marine mammal populations

Sound propagation in porous materials

• Porous materials, such as foams, fibrous materials, and granular media, are widely used in architectural acoustics for sound absorption and noise control
• Understanding sound propagation in porous materials is crucial for designing effective acoustic treatments and predicting their performance
• Porous material properties, sound absorption mechanisms, acoustic impedance, and modeling techniques are key concepts for optimizing the use of porous materials in architectural acoustics

Porous material properties

• Porosity: the ratio of the volume of voids to the total volume of the material
  • Higher porosity generally leads to better sound absorption
• Flow resistivity: the resistance to air flow through the material per unit thickness
  • Affects the sound absorption behavior and the frequency range of effectiveness
• Tortuosity: a measure of the complexity of the pore structure
  • Higher tortuosity leads to more complex sound propagation paths and increased absorption
• Pore size distribution: the range and proportion of pore sizes within the material
  • Affects the frequency-dependent absorption characteristics

Sound absorption mechanisms

• Viscous losses: caused by the friction between the oscillating air particles and the pore walls
  • Dominant at low frequencies and in materials with small pores
• Thermal losses: caused by the heat exchange between the air and the pore walls
  • Dominant at high frequencies and in materials with large pores
• Structural losses: caused by the vibration of the material skeleton
  • Significant in elastic materials, such as foams and fibrous materials

Acoustic impedance of porous materials

• The complex ratio of the sound pressure to the particle velocity at the surface of the material
• Determines the reflection and absorption characteristics of the material
• Depends on the material properties, frequency, and angle of incidence
• Can be measured using impedance tubes or predicted using models (e.g., Delany-Bazley, Johnson-Champoux-Allard)

Modeling sound propagation in porous media

• Empirical models: based on measured material properties and fitting parameters (e.g., Delany-Bazley, Miki)
  • Simple to use but limited in accuracy and applicability
• Phenomenological models: based on physical principles and measurable material properties (e.g., Johnson-Champoux-Allard, Biot)
  • More accurate and versatile but require more input parameters
• Numerical models: based on solving the governing equations of sound propagation in porous media (e.g., finite element method, boundary element method)
  • Most accurate and flexible but computationally intensive and require detailed material characterization

Sound propagation in multilayered media

• Multilayered structures, such as walls, floors, and acoustic panels, are common in architectural acoustics for sound insulation and noise control
• Understanding sound propagation in multilayered media is essential for designing effective sound insulation systems and predicting their performance
• Reflection and transmission at interfaces, transfer matrix method, modeling techniques, and applications in architectural acoustics are key concepts for optimizing the use of multilayered structures

Reflection and transmission at interfaces

• When a sound wave encounters an interface between two media, part of the energy is reflected, and part is transmitted
• The reflection and transmission coefficients depend on the acoustic impedances of the media and the angle of incidence
• Normal incidence: the sound wave is perpendicular to the interface
  • Reflection coefficient: $R = \frac{Z_2 - Z_1}{Z_2 + Z_1}$, where $Z_1$ and $Z_2$ are the acoustic impedances of the media
  • Transmission coefficient: $T = \frac{2 Z_2}{Z_2 + Z_1}$
• Oblique incidence: the sound wave is at an angle to the interface
  • Reflection and transmission coefficients depend on the angle of incidence and the wave types (longitudinal or transverse)
  • Calculated using Snell's law and the boundary conditions at the interface

Transfer matrix method

• A technique for modeling sound propagation in multilayered media
• Represents each layer by a transfer matrix that relates the sound pressure and particle velocity at the layer interfaces
• The overall transfer matrix of the multilayered system is obtained by multiplying the individual layer matrices
• Allows for the calculation of reflection and transmission coefficients, as well as the sound pressure and particle velocity at any point in the system

Modeling sound propagation in multilayered structures

• Analytical models: based on the transfer matrix method and the acoustic properties of the layers (e.g., impedance, propagation constant)
  • Suitable for simple geometries and homogeneous layers
• Numerical models: based on solving the governing equations of sound propagation in multilayered media (e.g., finite element method, boundary element method)
  • Suitable for complex geometries and inhomogeneous layers
• Statistical energy analysis (SEA): a high-frequency modeling approach that considers the average energy flow between subsystems
  • Suitable for complex systems with high modal densities and weak coupling between subsystems

Applications in architectural acoustics

• Sound insulation of walls and floors: designing multilayered structures to minimize sound transmission between spaces
  • Optimizing the layer properties, thicknesses, and sequence for maximum transmission loss
• Acoustic panels and absorbers: designing multilayered structures to absorb sound energy and control reverberation
  • Optimizing the layer properties and thicknesses for maximum absorption and desired frequency response
• Noise barriers: designing multilayered structures to reduce outdoor noise propagation
  • Optimizing the layer properties, thicknesses, and shape for maximum insertion loss

Diffraction and scattering of sound

• Diffraction and scattering are important phenomena in architectural acoustics that affect sound propagation and the acoustic environment
• Understanding diffraction and scattering is essential for predicting sound fields in complex environments and designing effective acoustic treatments
• Diffraction around obstacles, scattering from rough surfaces, and modeling techniques are key concepts for optimizing the acoustic performance of architectural spaces

Diffraction around obstacles

• Diffraction occurs when a sound wave encounters an obstacle or an opening comparable to or smaller than its wavelength
• The sound wave bends around the obstacle or spreads out after passing through the opening
• The extent of diffraction depends on the ratio of the wavelength to the obstacle or opening size (Fresnel number)
• Diffraction leads to sound reaching areas that are not in the direct line of sight of the source
• Important for predicting sound propagation around barriers, through apertures, and in urban environments

Scattering from rough surfaces

• Scattering occurs when a sound wave encounters a surface with irregularities comparable to or larger than its wavelength
• The incident sound energy is redistributed in various directions, depending on the surface roughness and the angle of incidence
• The extent of scattering depends on the ratio of the wavelength to the surface roughness (roughness parameter)
• Scattering leads to a more diffuse sound field and reduced specular reflections
• Important for predicting the acoustic behavior of surfaces, such as diffusers, absorbers, and room boundaries

Modeling diffraction and scattering effects

• Analytical models: based on the geometrical theory of diffraction (GTD) or the uniform theory of diffraction (UTD)
  • Suitable for simple geometries and high frequencies
  • Kirchhoff approximation: assumes the obstacle is large compared to the wavelength and has a smooth surface
  • Fresnel-Kirchhoff diffraction formula: calculates the diffracted field based on the source, obstacle, and receiver geometry
• Numerical models: based on solving the wave equation or its approximations (e.g., boundary element method, finite difference time domain)
  • Suitable for complex geometries and low to medium frequencies
  • Boundary element method (BEM): solves the wave equation on the surface of the obstacle or scatterer
  • Finite difference time domain (FDTD): solves the wave equation on a grid of points in the domain
• Hybrid models: combine analytical and numerical methods to balance accuracy and computational efficiency
  • Example: Kirchhoff-BEM model for diffraction around large obstacles with BEM for complex surfaces

Refraction and diffusion of sound

• Refraction and diffusion are important phenomena in architectural acoustics that affect sound propagation and the distribution of sound energy in a space
• Understanding refraction and diffusion is essential for predicting sound fields in non-uniform environments and designing spaces with desired acoustic characteristics
• Refraction due to temperature and wind gradients, sound diffusion in rooms, and the use of diffusers in architectural acoustics are key concepts for optimizing the acoustic performance of architectural spaces

Refraction due to temperature gradients

• Refraction occurs when a sound wave propagates through a medium with a non-uniform temperature distribution
• The speed of sound varies with temperature, causing the sound wave to bend towards regions of lower temperature (lower speed of sound)
• The extent of refraction depends on the magnitude and direction of the temperature gradient
• Important for predicting sound propagation in outdoor environments and large indoor spaces with temperature stratification

Refraction due to wind gradients

• Refraction also occurs when a sound wave propagates through a medium with a non-uniform wind velocity distribution
• The effective speed of sound is the sum of the sound speed and the wind velocity component in the direction of propagation
• The sound wave bends towards regions of lower effective sound speed (headwind) and away from regions of higher effective sound speed (tailwind)
• The extent of refraction depends on the magnitude and direction of the wind gradient
• Important for predicting sound propagation in outdoor environments and large indoor spaces with air flow

Sound diffusion in rooms

• Diffusion is the process of redistributing sound energy evenly in a space, both spatially and temporally
• A diffuse sound field is characterized by a uniform sound pressure level and a balanced distribution of sound energy in all directions
• Diffusion is achieved through the use of irregular room shapes, non-parallel surfaces, and diffusing elements (diffusers)
• A highly diffuse sound field is desirable in many acoustic spaces, such as concert halls and recording studios, to ensure a uniform listening experience and to minimize the effects of discrete reflections