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Acoustic wave

An acoustic wave is a mechanical disturbance that propagates through an elastic medium, such as a gas, liquid, or solid, by causing the particles of the medium to oscillate longitudinally, resulting in alternating regions of compression and rarefaction that transmit variations in pressure and density.^[1] These waves, which carry energy without net transport of matter, are fundamentally governed by the acoustic wave equation, derived from Newton's laws and the properties of the medium, typically expressed as \frac{\partial^2 \chi}{\partial x^2} = \frac{1}{c_s^2} \frac{\partial^2 \chi}{\partial t^2}, where \chi is the displacement and c_s is the speed of sound.^[1] The speed of acoustic waves depends on the medium's elasticity and density; in air at standard conditions (20°C and 1 atm), it is approximately 343 m/s, calculated as c_s = \sqrt{\gamma P / \rho}, where \gamma is the adiabatic index (about 1.4 for air), P is the equilibrium pressure, and \rho is the density.^[1] In denser media like water or steel, the speed increases significantly—to around 1480 m/s in water and 5000–6000 m/s in steel—due to stronger interparticle forces.^[2] Acoustic waves exhibit properties such as superposition, allowing multiple waves to interfere constructively or destructively, and their intensity diminishes with distance, following an inverse square law in three dimensions for spherical spreading.^[1] Unlike transverse electromagnetic waves, acoustic waves require a physical medium for propagation and cannot travel through a vacuum, limiting their range in space but enabling diverse applications on Earth.^[2] Key characteristics include frequency (determining pitch, typically 20 Hz to 20 kHz for human hearing), amplitude (related to loudness), and wavelength (\lambda = c_s / f), with higher frequencies experiencing greater absorption in air due to viscous and thermal losses.^[2] These waves underpin phenomena in acoustics, from everyday sound transmission to advanced technologies like ultrasound imaging, seismic exploration, and nondestructive testing in materials science.^[1]

Fundamentals

Definition

An acoustic wave is a mechanical disturbance that propagates through an elastic medium as a longitudinal pressure wave, in which the displacement of particles occurs parallel to the direction of wave propagation. This results in alternating regions of compression, where particles are closer together, and rarefaction, where they are farther apart, creating oscillations in pressure, density, and particle velocity.^[2]^[3] Unlike electromagnetic waves, which can propagate through a vacuum, acoustic waves require a physical medium—such as gases, liquids, or solids—for transmission, as they rely on the interactions between particles in the medium to carry the disturbance.^[4]^[5] These waves are generated by vibrating sources that perturb the medium, such as the prongs of a struck tuning fork or the diaphragm of a speaker, which produce successive compressions and rarefactions that travel outward from the source.^[6] The frequency of acoustic waves spans a broad range, with audible sound typically between 20 Hz and 20 kHz for human hearing, while frequencies below 20 Hz constitute infrasound and those above 20 kHz are ultrasound. The amplitude of an acoustic wave determines its intensity, which is proportional to the square of the amplitude, and relates to perceived loudness, where higher amplitudes correspond to greater loudness on a logarithmic scale.^[7]^[8]

Types

Acoustic waves are classified based on the medium of propagation, the direction of particle motion relative to the wave's direction, and frequency ranges. In fluids, such as air or water, acoustic waves are longitudinal, characterized by particle oscillations parallel to the direction of propagation, resulting in alternating compressions and rarefactions without any shear component.^[9] This type dominates in gaseous and liquid media due to the inability of fluids to support shear stresses.^[10] In solids, acoustic waves can be either longitudinal or transverse. Longitudinal waves in solids, often called compressional or P-waves, feature particle motion parallel to propagation, similar to those in fluids but capable of transmitting shear as well. Transverse shear waves, or S-waves, exhibit particle displacement perpendicular to the propagation direction, enabling polarization in two orthogonal planes and allowing propagation only in materials that resist shear, like solids.^[10] Surface waves, confined near boundaries, include Rayleigh waves, which combine longitudinal and vertical shear components to produce elliptical particle orbits that decay exponentially with depth from the surface, typically penetrating about one wavelength. Love waves, a type of transverse surface wave, involve horizontal particle motion perpendicular to propagation and are prominent in layered media, such as the Earth's crust.^[11]^[12] Acoustic waves are further categorized as bulk or guided. Bulk waves propagate freely through the interior of a homogeneous medium without boundary constraints, encompassing the longitudinal and transverse modes described above. Guided waves, in contrast, are confined within structures like waveguides, rods, or plates, where boundaries reflect and interfere with the waves to form discrete modes that propagate along the guide, often used in nondestructive testing for long-range inspection.^[13] Frequency-based classification delineates infrasonic waves (below 20 Hz), audible waves (20 Hz to 20 kHz), and ultrasonic waves (above 20 kHz). Infrasonic waves, generated by events like earthquakes or volcanic eruptions, are inaudible to humans but detectable by animals such as elephants. Audible waves correspond to the human hearing range, encompassing everyday sounds like speech. Ultrasonic waves find applications in medical imaging, such as echocardiography, where high-frequency pulses enable detailed visualization of tissues.^[14]^[15]^[16] In anisotropic media, such as crystals, acoustic waves exhibit polarization, where the direction of particle motion is constrained by the medium's directional elastic properties, leading to quasi-longitudinal (quasi-P) and quasi-transverse (quasi-S) waves with velocity and polarization dependent on propagation direction.^[17]

Mathematical Description

Wave Equation

The linear acoustic wave equation governs the propagation of small-amplitude pressure disturbances in fluids, serving as the foundational partial differential equation for acoustics. It arises from applying conservation principles to perturbed fluid states, yielding a second-order hyperbolic PDE that predicts wave behavior under idealized conditions.^[18] The derivation relies on key assumptions: the fluid is inviscid, meaning viscous effects are neglected in the momentum balance; the flow is irrotational, allowing representation via a scalar velocity potential; thermodynamic processes are adiabatic, linking pressure and density deviations through the speed of sound; and perturbations are small compared to equilibrium values, enabling linearization of nonlinear equations.^[19]^[20] These conditions approximate many practical scenarios in air or water where sound waves dominate.^[18] Consider a fluid at equilibrium with uniform pressure p_0, density \rho_0, and zero velocity. A small acoustic disturbance introduces deviations: pressure p = p_0 + p', density \rho = \rho_0 + \rho', and particle velocity \mathbf{u}. The linearized continuity equation, expressing mass conservation, is

\frac{\partial \rho'}{\partial t} + \rho_0 \nabla \cdot \mathbf{u} = 0.

The linearized momentum equation from Newton's second law (Euler form for inviscid flow) is

\rho_0 \frac{\partial \mathbf{u}}{\partial t} = -\nabla p'.

For adiabatic perturbations, the equation of state relates deviations via p' = c^2 \rho', where c = \sqrt{\gamma p_0 / \rho_0} is the speed of sound and \gamma is the adiabatic index.^[18] To obtain the wave equation, take the time derivative of the continuity equation:

\frac{\partial^2 \rho'}{\partial t^2} + \rho_0 \nabla \cdot \left( \frac{\partial \mathbf{u}}{\partial t} \right) = 0.

Substitute \partial \mathbf{u}/\partial t = -(1/\rho_0) \nabla p' = -(c^2 / \rho_0) \nabla \rho' from the momentum and state equations:

\frac{\partial^2 \rho'}{\partial t^2} = c^2 \nabla^2 \rho'.

Since p' = c^2 \rho', the equation for the acoustic pressure deviation p' (denoted simply as p hereafter) is the linear acoustic wave equation:

\begin{equation} \frac{\partial^2 p}{\partial t^2} = c^2 \nabla^2 p. \end{equation}

This scalar PDE holds in three dimensions and encapsulates wave propagation.^[18] Particular solutions illustrate wave forms satisfying the equation. Plane waves, representing wavefronts of constant phase on infinite planes, take the form p = A \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \phi), where A is amplitude, \mathbf{k} is the wave vector with magnitude k = \omega / c, \omega is angular frequency, \mathbf{r} is position, and \phi is phase.^[21] Spherical waves, approximating radiation from a point source, are p = \frac{A}{r} \cos(kr - \omega t + \phi), featuring geometric spreading inversely proportional to radial distance r.^[21] For unidirectional propagation, such as plane waves along the x-axis, the three-dimensional equation simplifies to the one-dimensional form:

\frac{\partial^2 p}{\partial t^2} = c^2 \frac{\partial^2 p}{\partial x^2},

whose general solution is p(x, t) = f(x - c t) + g(x + c t), combining arbitrary rightward and leftward traveling components.^[18]

Phase and Wavelength

Acoustic waves exhibit an oscillatory nature, typically described by sinusoidal functions that capture their periodic variations in pressure, density, or particle displacement. The phase of an acoustic wave refers to the argument of this sinusoidal function, often expressed as \phi = \omega t - \mathbf{k} \cdot \mathbf{r} + \phi_0, where \omega is the angular frequency, t is time, \mathbf{k} is the wave vector, \mathbf{r} is the position vector, and \phi_0 is a constant phase offset.^[22] This phase determines the wave's position within its cycle at any given point and time, reflecting the progression of the disturbance through the medium.^[23] The phase velocity v_p, which describes the speed at which a point of constant phase propagates, is given by v_p = \omega / k, where k = |\mathbf{k}| is the magnitude of the wave vector, known as the wave number.^[22] The wave number k quantifies the spatial frequency of the wave and is related to the wavelength \lambda by k = 2\pi / \lambda.^[24] Similarly, the angular frequency \omega measures the temporal rate of phase change and connects to the ordinary frequency f via \omega = 2\pi f.^[24] The wavelength \lambda represents the spatial periodicity of the wave, defined as the distance over which the wave's pattern repeats, and is expressed as \lambda = c / f, where c is the propagation speed of the wave.^[22] In acoustic wave trains, the phase difference between two waves or points along a wave is the separation in their phase arguments, which can lead to constructive or destructive interference depending on whether it corresponds to an integer or half-integer multiple of $2\pi, respectively.^[23] Coherence arises when this phase difference remains constant over time, allowing predictable interference patterns; incoherent waves, by contrast, exhibit randomly varying phase differences, resulting in no net interference.^[23] The temporal period T of an acoustic wave is the time for one complete oscillation, given by T = 1/f = 2\pi / \omega, while the spatial period is simply the wavelength \lambda = 2\pi / k, encapsulating the wave's repetitive behavior in both time and space.^[24] These parameters collectively define the oscillatory characteristics essential for understanding wave propagation and interaction in acoustic media.^[22]

Propagation Speed

The propagation speed of acoustic waves, often denoted as the speed of sound c, is a fundamental property that determines how quickly pressure disturbances travel through a medium. In ideal gases, this speed is given by the formula c = \sqrt{\frac{\gamma [P](/page/Pressure)}{\rho}}, where \gamma is the adiabatic index (ratio of specific heats), P is the pressure, and \rho is the density.^[25] This expression arises from the adiabatic compression and rarefaction of gas parcels during wave propagation, linking thermodynamic properties to wave dynamics.^[26] In liquids, which are nearly incompressible fluids, the speed simplifies to c = \sqrt{\frac{K}{\rho}}, with K representing the bulk modulus that measures the medium's resistance to uniform compression.^[27] For example, in water at room temperature, this yields speeds around 1480 m/s, significantly higher than in air due to the greater stiffness of liquids.^[27] Solids support both longitudinal and shear acoustic waves, with distinct speeds derived from the material's elastic constants. The longitudinal wave speed is c_L = \sqrt{\frac{\lambda + 2\mu}{\rho}}, where \lambda and \mu are the Lamé parameters characterizing interatomic interactions, while the shear wave speed is c_S = \sqrt{\frac{\mu}{\rho}}.^[28] These formulas reflect the coupling of compressional and shear deformations in solids, leading to typical values like c_L \approx 5000 m/s and c_S \approx 3000 m/s in metals such as steel.^[28] Environmental factors influence the speed in gases like air. Temperature has the dominant effect, with an approximate linear relation c \approx 331 + 0.6T m/s, where T is in degrees Celsius; this stems from the direct proportionality of c to the square root of absolute temperature in the ideal gas formula.^[29] Pressure has negligible impact at constant temperature, as c remains independent of P since density scales proportionally.^[30] Humidity slightly increases the speed; at 20°C, sound travels about 0.35% faster (approximately 1.2 m/s) in air at 100% relative humidity compared to dry air, due to the lower average molecular mass of moist air.^[31] In complex media such as crystals or composites, acoustic waves exhibit anisotropy, where propagation speed varies with direction relative to the material's symmetry axes, requiring tensorial descriptions of elastic properties beyond isotropic assumptions.^[32] Additionally, dispersion occurs in such media, causing the phase speed to depend on frequency owing to frequency-selective scattering or relaxation processes, which complicates wave propagation compared to non-dispersive cases.^[33]

Propagation Phenomena

Reflection and Transmission

When an acoustic wave encounters an interface between two different media, part of the wave is reflected back into the incident medium, while the remainder is transmitted into the second medium. This partitioning of energy is governed by the acoustic properties of the media, particularly their densities and sound speeds. The behavior arises from the requirement that certain physical quantities remain continuous across the boundary.^[34] The key boundary conditions at the interface are the continuity of acoustic pressure and the normal component of particle velocity. These conditions ensure that there is no abrupt change in pressure or net flow of mass across the boundary, leading to the generation of reflected and transmitted waves whose amplitudes satisfy these constraints.^[34] For normal incidence, where the wave approaches perpendicular to the interface, the reflection and transmission coefficients are determined by the acoustic impedances Z_1 = \rho_1 c_1 and Z_2 = \rho_2 c_2 of the two media, with \rho denoting density and c the speed of sound. The pressure reflection coefficient is given by

R = \frac{Z_2 - Z_1}{Z_2 + Z_1},

which represents the ratio of the reflected pressure amplitude to the incident pressure amplitude. The pressure transmission coefficient is

T = \frac{2 Z_2}{Z_2 + Z_1},

the ratio of the transmitted pressure amplitude to the incident. When Z_2 > Z_1, R is positive, indicating no phase change upon reflection; when Z_2 < Z_1, R is negative, signifying a 180-degree phase shift. These coefficients conserve energy, with the intensity reflection coefficient |R|^2 and transmission coefficient T_I = \frac{4 Z_1 Z_2}{(Z_1 + Z_2)^2} summing to unity.^[34] A classic example is an acoustic wave in air incident on a water surface, where Z_1 \approx 415 kg/m²s for air and Z_2 \approx 1.48 \times 10^6 kg/m²s for water. This yields R \approx 0.999, resulting in nearly total reflection and a faint echo, with only about 0.1% of the incident intensity transmitted into the water. Such high reflection at air-fluid interfaces is responsible for phenomena like underwater sound barriers and sonar echoes.^[34] At oblique incidence, where the wave strikes the interface at an angle, the situation is more complex, involving both reflection and refraction. The direction of the transmitted wave follows Snell's law:

\frac{\sin \theta_i}{\sin \theta_t} = \frac{c_1}{c_2},

where \theta_i is the incident angle, \theta_t the transmitted angle, and c_1, c_2 the sound speeds in the respective media. This law arises from the continuity of the tangential component of the wave vector across the boundary. The reflection coefficient for oblique incidence depends on the angle and impedances, generally reducing transmission as the angle increases.^[35] Total internal reflection occurs for oblique incidence when the wave propagates from a medium with lower sound speed (c_1 < c_2) to one with higher speed, and the incident angle exceeds the critical angle \theta_c = \sin^{-1}(c_1 / c_2). Beyond this angle, no energy transmits into the second medium; the wave is fully reflected, accompanied by an evanescent wave parallel to the interface. This effect, analogous to optics, enables applications like acoustic waveguides. Propagation speed differences between media drive the impedance mismatch underlying these reflections.^[36]

Refraction

Refraction of acoustic waves occurs when the speed of sound varies spatially within a medium, causing wave paths to bend toward regions of lower speed. This phenomenon arises in inhomogeneous media where gradients in sound speed c, often due to variations in temperature, density, or salinity, alter the direction of propagation. In the high-frequency limit, ray acoustics approximates this bending through the eikonal equation, which governs the travel time \tau along ray paths:

|\nabla \tau| = \frac{1}{c}

Here, \nabla \tau represents the gradient of the travel time, and rays follow paths orthogonal to the wavefronts, curving according to local speed variations.^[37] In stratified media with continuous speed gradients, such as oceanic layers, refraction can produce acoustic mirages and shadowing zones. Acoustic mirages emerge from strong near-surface temperature gradients, like those above a hot roadway, where sound speed increases rapidly with height, bending rays upward and creating illusory distant sources or excess attenuation of 10–20 dB in shadow zones for frequencies of 2–10 kHz. In the ocean, the sound channel (or SOFAR channel) forms a waveguide due to downward refraction in the thermocline, trapping low-frequency rays and enabling long-range propagation while producing shadow zones beyond the channel where rays are refracted away from receivers.^[38]^[39]^[40] Ray paths in these media adhere to Fermat's principle, which states that acoustic rays follow stationary travel time paths—typically minima, maxima, or saddle points—between source and receiver. This variational principle ensures that, in inhomogeneous fields like varying air temperature, the eigenray minimizes or extremizes the integral \tau = \int ds / c along the path, with numerical methods confirming identical trajectories across variational, Hamiltonian, and geometric approaches.^[41] In graded-index media, where sound speed varies continuously across the propagation direction, refraction leads to beam displacement and focusing effects. Beams experience lateral shifts as rays curve through speed gradients, analogous to optical beam displacement at interfaces but extended to continuous media. Gradient-index acoustic metasurfaces, engineered with subwavelength structures like porous silicone rubber, enable precise focusing of airborne sound at scales below the wavelength, achieving subwavelength resolution for imaging by modulating phase and redirecting rays toward a focal point.^[42] Atmospheric propagation exemplifies these effects through temperature inversions, where sound speed increases with height, refracting rays downward and extending audible range, as in hearing distant sounds across water at night. Conversely, daytime lapse rates bend rays upward, forming shadow zones that limit propagation.^[43]

Absorption and Attenuation

Absorption and attenuation refer to the irreversible loss of energy in acoustic waves as they propagate through a medium, converting mechanical energy into heat via dissipative mechanisms. These processes reduce the amplitude of the wave over distance, distinguishing them from reversible phenomena like reflection. In acoustic propagation, attenuation is quantified by the coefficient α, where the pressure amplitude decays as e^{-αx}, leading to intensity I decaying as I = I_0 e^{-2αx}, with x the propagation distance.^[44] Classical absorption arises from viscous friction and thermal conduction, the primary dissipative effects in fluids for low-amplitude waves. Viscous losses stem from shear and bulk viscosity resisting particle motion, while thermal conduction causes heat transfer across temperature gradients induced by compression and rarefaction. The combined attenuation coefficient for these processes, known as the Stokes-Kirchhoff formula, is given by

\alpha = \frac{\omega^2}{2 \rho c^3} \left( \frac{4}{3} \eta + \frac{(\gamma - 1) \kappa}{C_p} \right),

where ω is the angular frequency, ρ the density, c the speed of sound, η the shear viscosity, κ the thermal conductivity, γ the adiabatic index, and C_p the specific heat at constant pressure.^[45]^[44] This expression shows classical absorption scales quadratically with frequency (∝ ω²), a dependence prominent in ultrasound applications where higher frequencies lead to greater attenuation per unit distance.^[46] Beyond classical mechanisms, relaxation processes contribute significantly to attenuation, particularly in gases and liquids where internal molecular degrees of freedom lag behind the rapid pressure changes of the wave. These include vibrational relaxation, where energy is temporarily stored in molecular vibrations before dissipating as heat, and rotational relaxation in polyatomic gases. Chemical relaxation occurs in reacting mixtures, such as seawater where magnesium sulfate dissociation absorbs sound via equilibrium shifts.^[44]^[47] Such processes introduce frequency-dependent peaks in attenuation, contrasting the smooth quadratic rise of classical absorption. At high amplitudes, nonlinear effects amplify attenuation through waveform distortion. Finite amplitude propagation steepens the wave profile, generating higher harmonics that increase energy dissipation via classical and relaxation mechanisms. Eventually, this leads to shock formation, where the waveform develops a sharp discontinuity, further enhancing absorption and limiting propagation distance in applications like sonic booms.^[48]^[49]

Wave Interactions

Interference

Acoustic waves, being linear, obey the principle of superposition, whereby the total displacement at any point in the medium is the vector sum of the displacements produced by each individual wave.^[1] This principle holds because acoustic waves in fluids and solids typically involve small-amplitude perturbations where nonlinear effects are negligible.^[50] When two acoustic waves of slightly different frequencies, f_1 and f_2, superpose, they produce an interference pattern known as beats, characterized by periodic variations in amplitude at a modulation frequency of |f_1 - f_2|.^[51] The resulting sound intensity fluctuates with a period equal to $1/|f_1 - f_2|, creating an audible pulsing effect that is most prominent when the frequency difference is small, such as a few hertz.^[52] An analogy to Young's double-slit experiment in optics can be drawn for sound waves using two coherent point sources, such as speakers, separated by a small distance. Constructive interference occurs at points where the path length difference \delta from the two sources satisfies \delta = m\lambda, with m an integer and \lambda the wavelength, leading to maxima in sound intensity.^[53] This setup produces observable fringes of louder and quieter regions, demonstrating spatial interference patterns in air.^[54] For such interference fringes to be clearly observable, the waves must maintain coherence over the relevant path lengths, quantified by the coherence length, which is the distance beyond which phase relationships become random due to factors like source bandwidth or medium fluctuations.^[55] In acoustics, coherence lengths on the order of meters are typical for narrowband sources in controlled environments, but they decrease with broader frequency spectra or turbulent propagation. A practical application of destructive interference is found in noise-canceling headphones, where microphones detect ambient noise and generate an antiphase signal that superposes with the incoming waves to attenuate low-frequency sounds through cancellation.^[56] This technique relies on the superposition principle to achieve reductions of up to 20-30 dB in targeted frequency bands, primarily below 1 kHz.

Diffraction

Diffraction refers to the bending and spreading of acoustic waves as they encounter obstacles or pass through apertures, allowing sound to propagate into regions that would otherwise be shadowed in geometric acoustics. This phenomenon arises because acoustic waves, like other waves, do not strictly follow straight-line paths but instead curve around edges due to their wave nature. The extent of diffraction depends on the wavelength relative to the obstacle size; longer wavelengths diffract more readily than shorter ones.^[57] The Huygens-Fresnel principle provides the foundational explanation for acoustic diffraction, positing that every point on a wavefront acts as a source of secondary spherical wavelets that propagate forward and interfere to form the subsequent wavefront. In acoustics, this principle models how plane waves bend around barriers or through slits by treating the wavefront as a superposition of these wavelets, with diffraction patterns emerging from their coherent summation. For instance, in surface acoustic wave propagation near a channel wall, the principle predicts a knife-edge diffraction effect where the wave interacts with the boundary to produce cylindrical wavefronts.^[58]^[59] A classic demonstration of acoustic diffraction occurs in the single-slit configuration, where a plane wave passes through a narrow aperture of width a. The resulting intensity pattern features a central maximum flanked by minima, located at angles \theta satisfying \sin \theta = m \lambda / a, where m = \pm 1, \pm 2, \dots is the order and \lambda is the wavelength. This pattern arises from the destructive interference of wavelets from different parts of the slit, with the first minimum (m = \pm 1) marking the edge of the central lobe. Numerical simulations confirm that this behavior holds in acoustic media, analogous to optics but influenced by the medium's speed of sound.^[60] Acoustic diffraction patterns are classified into near-field (Fresnel) and far-field (Fraunhofer) regimes based on the observation distance relative to the aperture size and wavelength. In the Fresnel regime, close to the aperture, the pattern evolves with quadratic phase terms, capturing curvature effects and producing complex fringes that depend on distance. The Fraunhofer regime, at greater distances where the wavefront approximates a plane, yields simpler patterns independent of position, often analyzed via Fourier transforms of the aperture function. The transition occurs roughly when the distance z exceeds a^2 / \lambda, with Fresnel diffraction dominating in shadowed regions behind barriers.^[57] In acoustic imaging applications, diffraction imposes fundamental resolution limits, quantified by the Rayleigh criterion, which states that two point sources are resolvable if separated by at least \theta \approx 1.22 \lambda / D, where D is the aperture diameter. For ultrasound imaging, this yields a lateral resolution on the order of the wavelength, typically limiting super-resolution techniques without advanced methods like structured illumination. Exceeding this limit requires compensating for diffracted wavefronts to sharpen images beyond the diffraction barrier.^[61] Practical examples illustrate diffraction's role in sound propagation. In sound localization, diffraction around obstacles like head or barriers provides cues for non-line-of-sight sources; for instance, edge diffraction from a plate can shift perceived horizontal position by 1–3° at mid-frequencies (500–2000 Hz), aiding human or algorithmic detection in occluded environments. Similarly, noise barriers attenuate sound via diffraction over their tops, with excess attenuation typically 5–15 dB depending on frequency and geometry; low frequencies (<500 Hz) diffract more, reducing effectiveness, while optimized edges enhance insertion loss by 2–5 dB through modified Fresnel zones.^[62]^[63]

Standing Waves

Standing waves, also known as stationary waves, arise in acoustic systems when two coherent waves of identical frequency and amplitude travel in opposite directions along the same path, typically due to reflections from boundaries in a confined medium such as a pipe or tube. This superposition results in constructive interference at specific points called antinodes, where displacement amplitude is maximum, and destructive interference at nodes, where displacement is zero; these positions remain fixed relative to the medium.^[64] In longitudinal acoustic waves, nodes correspond to points of minimum pressure variation, while antinodes exhibit maximum pressure fluctuations, enabling the wave pattern to persist without net propagation. In acoustic pipes, boundary conditions dictate the allowable standing wave patterns and their frequencies. For a pipe closed at one end and open at the other, the closed end enforces a displacement node (pressure antinode), while the open end allows a displacement antinode (pressure node); the fundamental resonance frequency is thus f_1 = \frac{c}{4L}, where c is the speed of sound in the medium and L is the pipe length, with higher harmonics occurring at odd multiples: f_n = (2n-1) f_1 for n = 1, 2, [3, \dots](/page/3_Dots).^[65] Conversely, a pipe open at both ends supports antinodes at each extremity, yielding a fundamental frequency of f_1 = \frac{c}{2L} and all integer harmonics: f_n = n f_1 for n = 1, 2, [3, \dots](/page/3_Dots). These resonance frequencies represent the natural modes where minimal driving force sustains large-amplitude oscillations, as the system's impedance matches the source efficiently.^[66] The quality factor Q quantifies the resonance sharpness in such systems, defined as Q = 2\pi \times \frac{E_\text{stored}}{E_\text{lost}}, where E_\text{stored} is the peak energy in the mode and E_\text{lost} is the energy dissipated per oscillation cycle, often due to viscous drag and thermal effects in the fluid; higher Q values indicate narrower bandwidths around the resonance frequency and better energy retention.^[67] Standing waves store acoustic energy alternately as kinetic energy in particle motion near displacement antinodes and as potential (compressive) energy near pressure antinodes, with the total energy oscillating but remaining localized within the confined space.^[68] To maintain these oscillations against dissipative losses, an external driving source—such as airflow or a speaker—must supply energy at the resonance frequency, preventing amplitude decay and enabling sustained vibration.^[69] Practical examples of acoustic standing waves include organ pipes, where closed-open configurations produce rich timbres from odd harmonics, allowing precise pitch control via pipe length adjustments in musical performance. Similarly, the human vocal tract functions as a variable resonator, approximately 17 cm long and open at the mouth, supporting standing wave resonances (formants) that amplify specific harmonics from the glottal source to form distinct vowel sounds.

Applications

In Fluids

Acoustic waves propagate primarily as longitudinal waves in fluids, where particle motion aligns with the direction of wave travel, enabling efficient transmission through gases and liquids.^[70] In underwater acoustics, these waves facilitate sonar ranging systems that emit sound pulses to measure distances to submerged objects by calculating the time for echoes to return, essential for navigation and detection in opaque marine environments.^[70] Marine mammals, such as dolphins and whales, employ echolocation—a biological form of sonar—by generating high-frequency clicks that reflect off prey or obstacles, allowing precise localization in water where visibility is limited.^[71] This process relies on the relatively low attenuation of sound in seawater compared to light, enabling ranges up to several kilometers for certain species.^[72] Atmospheric applications harness low-frequency infrasound waves, below 20 Hz, for weather monitoring by detecting pressure perturbations from events like storms or tornadoes, which propagate over long distances with minimal dissipation in air.^[73] In aviation, acoustic waves inform noise control strategies, such as designing engine components to disrupt turbulent airflow that generates excessive sound, reducing community exposure near airports.^[74] High-intensity ultrasound in fluids introduces challenges like cavitation, where rapid pressure cycles form and collapse vapor bubbles, potentially causing tissue damage or enhancing processes like emulsification in medical and industrial settings.^[75] Acoustic radiation pressure, arising from the nonlinear interaction of intense sound waves with fluid particles, enables levitation and manipulation of small objects or droplets in air or liquids, suspending them against gravity without physical contact for applications in microgravity research.^[76] In medicine, ultrasound waves traverse water-based tissues, such as those in the human body, to produce diagnostic images by reflecting off boundaries between soft structures, providing non-invasive visualization of organs and blood flow.^[77]

In Solids

Acoustic waves in solids differ from those in fluids due to the ability of solids to support both longitudinal (compressional) and shear (transverse) wave modes, enabling a wider range of propagation behaviors influenced by material elasticity and anisotropy. These waves are generated and detected using piezoelectric transducers, which convert electrical energy into mechanical vibrations via the piezoelectric effect in materials like quartz or lead zirconate titanate (PZT), and vice versa for reception. This bidirectional transduction is fundamental to applications in solid media, where waves propagate at speeds typically ranging from 1 to 6 km/s depending on the material, such as 5.9 km/s for longitudinal waves in steel. Ultrasonic testing employs high-frequency acoustic waves (above 20 kHz) for non-destructive evaluation of solid structures, particularly welds and materials, using the pulse-echo technique. In this method, a short ultrasonic pulse is transmitted into the solid, and echoes from internal flaws like cracks or voids are received and analyzed to determine defect location and size based on time-of-flight measurements. Developed extensively since the mid-20th century, this technique is standardized in industries such as aerospace and manufacturing, with pulse-echo enabling detection of defects as small as 0.5 mm in metals. In geophysics, seismic waves—primary (P-waves, longitudinal) and secondary (S-waves, shear)—propagate through Earth's solid crust and mantle, providing critical data for earthquake modeling and subsurface imaging. P-waves travel faster (up to 8 km/s in the crust) and can pass through fluids, while S-waves, limited to solids, move at about 4-5 km/s and reveal shear modulus variations. These waves, generated by natural earthquakes or controlled sources, are modeled using elastic wave equations to simulate rupture dynamics and predict ground motion, as in finite-difference simulations that incorporate attenuation and anisotropy. Guided acoustic waves, such as Lamb waves in thin plates and rods, are particularly useful for long-range inspection in solids due to their multimodal propagation confined by boundaries. Lamb waves, symmetric and antisymmetric modes, disperse with frequency and thickness, allowing sensitivity to defects at various depths; for instance, the A0 mode is effective for detecting surface cracks in aluminum plates up to several meters away. In rods, torsional and extensional guided waves similarly enable efficient energy transmission over distances, minimizing attenuation compared to bulk waves. A key application is structural health monitoring in bridges, where piezoelectric transducers are embedded or surface-mounted to continuously generate and detect guided waves for real-time flaw detection. For example, in steel girder bridges, Lamb wave arrays identify corrosion or fatigue cracks by analyzing changes in wave scattering patterns, with systems achieving detection ranges of 10-20 meters and accuracies better than 5% in damage localization. This approach enhances safety by enabling early intervention without disassembly.

Sensing and Imaging

Acoustic waves play a crucial role in sensing and imaging technologies by enabling the detection, measurement, and visualization of physical phenomena through pressure variations in media. Acoustic sensors convert these mechanical waves into electrical signals, facilitating applications from sound recording to underwater exploration. In imaging, acoustic waves allow non-invasive probing of internal structures by exploiting wave propagation, reflection, and scattering properties. Microphones serve as fundamental acoustic sensors, typically employing a diaphragm that responds to pressure changes from incident sound waves, which deflects to modulate an electrical output via capacitive, piezoelectric, or resistive mechanisms. For instance, condenser microphones use a charged diaphragm and backplate to produce a voltage proportional to acoustic pressure, offering high sensitivity for audio applications. Hydrophones, designed for underwater environments, similarly detect pressure waves but are optimized for low-frequency, high-pressure aquatic propagation, often using piezoelectric ceramics like lead zirconate titanate (PZT) to transduce signals in sonar systems. These sensors are essential for marine mammal monitoring and submarine detection, with sensitivities reaching -200 dB re 1 V/μPa. Pulse-echo ultrasound represents a primary imaging modality, where short acoustic pulses are transmitted into a medium, and echoes from interfaces or scatterers are received to reconstruct spatial information, forming B-mode scans that display amplitude as grayscale intensity. This technique operates at frequencies from 1 to 20 MHz in medical contexts, achieving resolutions down to 0.1 mm for soft tissue visualization. Doppler ultrasound extends this by measuring frequency shifts in backscattered waves to quantify motion, such as blood flow velocity, calculated as v = \frac{(f_r - f_0) c}{2 f_0 \cos \theta}, where f_r is the received frequency, f_0 the transmitted frequency, c the speed of sound, and \theta the angle of insonation; this enables real-time vascular assessments with accuracies better than 10% for peak velocities up to 2 m/s. Photoacoustic imaging combines optical excitation with acoustic detection, wherein laser pulses absorbed by tissues generate thermoelastic waves that propagate and are captured to map optical absorption, providing high-contrast images of hemoglobin distribution in oncology at depths up to 5 cm. Array processing enhances acoustic sensing and imaging through beamforming, where multiple transducers are arranged to steer and focus waves directionally, improving signal-to-noise ratios by 10-20 dB in noisy environments. In phased-array ultrasound probes, time delays are applied to elements to form synthetic apertures, enabling 3D volumetric imaging with frame rates exceeding 50 Hz. Practical examples include fetal ultrasound, which uses 2-5 MHz probes for real-time B-mode and Doppler assessment of cardiac activity and amniotic fluid, detecting anomalies with over 95% sensitivity in routine obstetrics. In industrial settings, acoustic wave imaging via ultrasonic testing identifies flaws like cracks in metals, employing pulse-echo from arrays to achieve defect resolutions of 1 mm in weld inspections, critical for aerospace safety.

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