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Acoustic waves are types ofwaves that propagate through matter —such asgas,liquid, and/orsolids— by causing the particles of the medium to compress and expand. These waves carry energy and are characterized by properties likeacoustic pressure,particle velocity, andacoustic intensity. The speed of an acoustic wave depends on the properties of the medium it travels through; for example, it travels at approximately 343 meters per second in air, and 1480 meters per second in water. Acoustic waves encompass a broad range of phenomena, from audible sound to seismic waves and ultrasound, finding applications in diverse fields likeacoustics,engineering, andmedicine.

An acoustic wave is a mechanical wave that transmits energy through the movements of atoms and molecules. Acoustic waves transmit through fluids in alongitudinal manner (movement of particles are parallel to the direction of propagation of the wave); in contrast to electromagnetic waves that transmit intransverse manner (movement of particles at a right angle to the direction of propagation of the wave). However, in solids, acoustic waves transmit in both longitudinal and transverse manners due to presence ofshear moduli in such a state of matter.^[1]

Theacoustic wave equation describes the propagation of sound waves. The acoustic wave equation forsound pressure in onedimension is given by$${\displaystyle \frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }{x}^2}-\frac{1}{c^2}\frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }{t}^2}=0}$$where

• ${\displaystyle p}$ issound pressure inPa
• ${\displaystyle x}$ is position in the direction of propagation of the wave, inm
• ${\displaystyle c}$ isspeed of sound inm/s
• ${\displaystyle t}$ istime ins

The wave equation forparticle velocity has the same shape and is given by$${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}-\frac{1}{c^2}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}=0}$$where

• ${\displaystyle u}$ isparticle velocity inm/s

For lossy media, more intricate models need to be applied in order to take into account frequency-dependent attenuation and phase speed. Such models include acoustic wave equations that incorporate fractional derivative terms, see also theacoustic attenuation article.

D'Alembert gave the general solution for the lossless wave equation. For sound pressure, a solution would be$${\displaystyle p=R\cos (\omega t-kx)+(1-R)\cos (\omega t+kx)}$$where

• ${\displaystyle \omega }$ isangular frequency in rad/s
• ${\displaystyle t}$ is time ins
• ${\displaystyle k}$ iswave number in rad·m⁻¹
• ${\displaystyle R}$ is a coefficient without unit

For${\displaystyle R=1}$ the wave becomes a travelling wave moving rightwards, for${\displaystyle R=0}$ the wave becomes a travelling wave moving leftwards. Astanding wave can be obtained by${\displaystyle R=0.5}$.

In a travelling wave pressure and particle velocity are inphase, which means the phase angle between the two quantities is zero.

This can be easily proven using theideal gas law$${\displaystyle pV=nRT}$$where

• ${\displaystyle p}$ ispressure inPa
• ${\displaystyle V}$ is volume in m³
• ${\displaystyle n}$ is amount inmol
• ${\displaystyle R}$ is theuniversal gas constant with value$8.314472(15)\frac{\mathrm{J}}{\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{K}}$

Consider a volume${\displaystyle V}$. As an acoustic wave propagates through the volume, adiabatic compression and decompression occurs. For adiabatic change the following relation between volume${\displaystyle V}$ of a parcel of fluid and pressure${\displaystyle p}$ holds$${\displaystyle \frac{\mathrm{\partial }V}{V_m}=\frac{-1}{\gamma }\frac{\mathrm{\partial }p}{p_m}}$$where${\displaystyle \gamma }$ is theadiabatic index without unit and the subscript${\displaystyle m}$ denotes the mean value of the respective variable.

As a sound wave propagates through a volume, the horizontal displacement of a particle${\displaystyle \eta }$ occurs along the wave propagation direction.$${\displaystyle \frac{\mathrm{\partial }\eta }{V_m}A=\frac{\mathrm{\partial }V}{V_m}=\frac{-1}{\gamma }\frac{\mathrm{\partial }p}{p_m}}$$where

• ${\displaystyle A}$ is cross-sectional area in m²

From this equation it can be seen that when pressure is at its maximum, particle displacement from average position reaches zero. As mentioned before, the oscillating pressure for a rightward traveling wave can be given by$${\displaystyle p=p_0\cos (\omega t-kx)}$$Since displacement is maximum when pressure is zero there is a 90 degrees phase difference, so displacement is given by$${\displaystyle \eta ={\eta }_0\sin (\omega t-kx)}$$Particle velocity is the first derivative of particle displacement:${\displaystyle u=\mathrm{\partial }\eta /\mathrm{\partial }t}$. Differentiation of a sine gives a cosine again$${\displaystyle u=u_0\cos (\omega t-kx)}$$

During adiabatic change, temperature changes with pressure as well following$${\displaystyle \frac{\mathrm{\partial }T}{T_m}=\frac{\gamma -1}{\gamma }\frac{\mathrm{\partial }p}{p_m}}$$This fact is exploited within the field ofthermoacoustics.

The propagation speed, or acoustic velocity, of acoustic waves is a function of the medium of propagation. In general, the acoustic velocityc is given by the Newton-Laplace equation:$${\displaystyle c=\sqrt{\frac{C}{\rho }}}$$where

• C is acoefficient of stiffness, thebulk modulus (or the modulus of bulk elasticity for gas mediums),
• ${\displaystyle \rho }$ is thedensity in kg/m³

Thus the acoustic velocity increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of the material, and decreases with the density.For general equations of state, if classical mechanics is used, the acoustic velocity${\displaystyle c}$ is given by$${\displaystyle c^2=\frac{\mathrm{\partial }p}{\mathrm{\partial }\rho }}$$with${\displaystyle p}$ as the pressure and${\displaystyle \rho }$ the density, where differentiation is taken with respect to adiabatic change.

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Acoustic waves are elastic waves that exhibit phenomena likediffraction,reflection andinterference. Note thatsound waves in air are notpolarized since they oscillate along the same direction as they move.

Interference is the addition of two or more waves that results in a new wave pattern. Interference of sound waves can be observed when two loudspeakers transmit the same signal. At certain locations constructive interference occurs, doubling the local sound pressure. And at other locations destructive interference occurs, causing a local sound pressure of zero pascals.

Astanding wave is a special kind of wave that can occur in aresonator. In a resonatorsuperposition of the incident and reflective wave occurs, causing a standing wave. Pressure and particle velocity are 90 degrees out of phase in a standing wave.

Consider a tube with two closed ends acting as a resonator. The resonator hasnormal modes at frequencies given by$${\displaystyle f=\frac{Nc}{2d}N\in \{1,2,3,\dots \}}$$where

• ${\displaystyle c}$ is the speed of sound inm/s
• ${\displaystyle d}$ is the length of the tube inm

At the ends particle velocity becomes zero since there can be no particle displacement. Pressure however doubles at the ends because of interference of the incident wave with the reflective wave. As pressure is maximum at the ends while velocity is zero, there is a 90 degrees phase difference between them.

An acoustic travelling wave can bereflected by a solid surface. If a travelling wave is reflected, the reflected wave can interfere with the incident wave causing a standing wave in thenear field. As a consequence, the local pressure in the near field is doubled, and the particle velocity becomes zero.

Attenuation causes the reflected wave to decrease in power as distance from the reflective material increases. As the power of the reflective wave decreases compared to the power of the incident wave, interference also decreases. And as interference decreases, so does the phase difference between sound pressure and particle velocity. At a large enough distance from the reflective material, there is no interference left anymore. At this distance one can speak of thefar field.

The amount of reflection is given by the reflection coefficient which is the ratio of the reflected intensity over the incident intensity$${\displaystyle R=\frac{I_{\text{reflected}}}{I_{\text{incident}}}}$$

Acoustic waves can be absorbed. The amount of absorption is given by the absorption coefficient which is given by$${\displaystyle \alpha =1-R^2}$$where

• ${\displaystyle \alpha }$ is theabsorption coefficient without a unit
• ${\displaystyle R}$ is thereflection coefficient without a unit

Oftenacoustic absorption of materials is given in decibels instead.

When an acoustic wave propagates through a non-homogeneous medium, it will undergo diffraction at the impurities it encounters or at the interfaces betweenlayers of different materials. This is a phenomenon very similar to that of the refraction, absorption and transmission oflight inBragg mirrors. The concept of acoustic wave propagation through periodic media is exploited with great success inacoustic metamaterial engineering.^[2]

The acoustic absorption, reflection and transmission in multilayer materials can be calculated with thetransfer-matrix method.^[3]

• Wave mechanics
• Acoustics
• Sound

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