Acoustics

Acoustics studies sound: its generation, propagation, interaction with matter and detection. In a fluid, a small acoustic wave couples a pressure disturbance to the motion of the medium. The disturbance transports energy without requiring the same parcel of material to travel from the source to the receiver.

A loudspeaker and a microphone make a simple experiment. Changing their separation measures propagation delay, while changing the frequency in a bounded room reveals resonance and standing waves. A strong microphone response does not by itself imply a large net sound intensity: oppositely travelling waves can produce a large local pressure while their mean energy transport cancels. Acoustic impedance connects pressure and motion for a specified travelling wave.

KEMAR, the Knowles Electronics Manikin for Acoustic Research, is a head-and-torso acoustic simulator used to measure how sound interacts with an approximately human-sized listening geometry. Its external ears, ear-canal system and acoustic loads make it more informative for many hearing-device tests than a microphone placed alone in free space. The manufacturer's description specifies representative adult dimensions and ear-simulation properties. ^[ref]

An original manikin paper by M. D. Burkhard and R. M. Sachs appeared in 1975. Later studies used KEMAR to investigate device coupling as well as the surrounding sound field.^[ref]

For example, Abhijit Kulkarni and H. Steven Colburn repeatedly placed headphone cushions on a KEMAR and measured the resulting headphone transfer function. Repositioning changed the response, complicating compensation by a single fixed inverse filter. The experiment demonstrates that the ear–device interface is part of the acoustic system, rather than an incidental mounting detail.^[ref]

An invented measurement comparison makes the meaning concrete. If the same electrical drive produces pressures of 0.020 and 0.010 pascal at one measurement frequency in two fittings, their sound pressure levels differ by $20{\log }_{10}2\approx 6.02$ decibels. This is a fit-dependent response difference at a specified point, not an automatic judgment of perceived loudness or device quality.

A standardized manikin supports reproducible comparisons but does not reproduce every listener's ear geometry, physiology or perception. Conclusions should specify the simulator configuration and measurement conditions.

An electroacoustic transducer converts between an electrical signal and acoustic motion. A loudspeaker or a pair of headphones converts electrical input into sound; a microphone converts acoustic input into an electrical output. The name describes the conversion role, not one universal physical mechanism.

In an idealized moving-coil device, a current-carrying conductor in a magnetic field experiences a force. If the active conductor length is $\ell$ and the relevant field strength is $B$, the perpendicular-component model gives $F=B\ell I$. With $B\ell =2N/A$ and current $I=0.020\mathrm{A}$, the force is $0.040\mathrm{N}$. A diaphragm and its mechanical load turn that force into motion and sound pressure; force alone does not determine the complete acoustic output.

Conversion therefore has several linked stages: electrical drive, mechanical motion and acoustic coupling. A pressure transfer function of the complete device can depend on frequency and its surroundings. Headphone measurements on an acoustic mannequin, for example, varied when the cushions were repositioned, despite using the same nominal transducer. This demonstrates a change in the coupled system rather than requiring a change in the electrical recording itself.^[ref]

The distinction is useful in calibration. A quoted electrical sensitivity, mechanical displacement or acoustic pressure is meaningful only with its input, measurement point and loading conditions specified. An electroacoustic transducer transfers energy and information through a physical interface; its useful behavior belongs to the device together with that interface.

Headphones are electroacoustic transducers worn at or near the ears that convert an electrical signal into local acoustic pressure variations. A two-channel pair can deliver different signals to the two ears. The relevant output is the sound pressure reaching the ear, which depends on the transducer, its placement and the surrounding acoustic geometry, not just the electrical input.

For a linear small-signal description, let $V(f)$ be the input electric potential difference and $p(f)$ the complex pressure amplitude at a specified ear location. A headphone transfer function $H(f)=p(f)/V(f)$ describes both amplitude and phase. Two headphones can receive the same $V(f)$ yet produce different $p(f)$; even the same pair can produce different responses after being repositioned.^[ref]

Abhijit Kulkarni and H. Steven Colburn measured responses on a KEMAR acoustic mannequin with repeated placements of the headphone cushions. Placement-dependent variation complicated compensation by a fixed inverse filter. This experiment shows why a measurement needs a defined coupling arrangement: moving the device can change the acoustic system being calibrated.^[ref]

For an illustrative amplitude example, if pressure doubles at a particular frequency while the reference pressure is unchanged, sound pressure level increases by $20{\log }_{10}2\approx 6.02$ decibels. That is a statement about pressure at the measurement point; it is not automatically a doubling of perceived loudness or a uniform change across frequencies. Headphone reproduction is therefore an electrical-to-mechanical-to-acoustic chain that must be assessed at its intended output location.

A headphone transfer function describes how an input electrical signal becomes acoustic pressure at a specified point near or within the listener's ear. For a linear small-signal description,where $p(f)$ is complex sound pressure amplitude and $V(f)$ the input electric potential difference. This transfer function includes both magnitude and phase. With these variables its units are pascals per volt, rather than a dimensionless gain.^[ref]

For example, at one frequency let $H=2Pa/V$ and apply a $10\mathrm{mV}$ sinusoidal input. The pressure amplitude is $0.020\mathrm{Pa}$ under the model. A compensating filter for a desired pressure $p_d$ would ideally supply $V=p_d/H$, but division by a small or uncertain response can demand a large, fragile correction.

The coupling to the ear is part of the system. Abhijit Kulkarni and H. Steven Colburn repeatedly positioned headphone cushions on a KEMAR acoustic mannequin and measured changes in response. Their measurements showed that repositioning could complicate compensation using a fixed inverse filter. A transfer function measured for one fitting need not remain identical after the device is moved.^[ref]

This differs from a head-related transfer function, which characterizes transformation from an external sound source to an ear location. Reproducing a desired spatial sound over headphones can require accounting for both transformations, rather than confusing the headphone's coloration with the directional effect being simulated.

Thermoacoustics studies coupling between thermal processes and acoustic motion. Oscillatory compression changes gas temperature; heat exchange at the right phase can amplify a sound wave or let sound transport heat.

In an engine, a maintained temperature difference can drive acoustic oscillations. In a refrigerator, externally supplied acoustic work can pump heat. The timing and location of heat exchange determine the energy transfer, so a temperature gradient alone does not guarantee self-sustained sound. Viscous and thermal boundary layers are important near the heat-exchanging structures.

Distribution of the components of sound waves over frequency. The relative amplitudes and their evolution contribute to timbre; a spectrum is distinct from a statement of the fundamental pitch alone.

A free reed aerophone produces sound when airflow drives an elastic tongue that moves through a closely fitting opening. Pressure and reed motion interact, repeatedly changing the available flow area and sustaining oscillation.

The reed's inertia and effective spring constant set a mechanical frequency scale; coupling to the surrounding air changes the actual oscillation and its onset. A pressure-driven reed mounted over a small chamber provides a concrete experiment: optical motion measurements can be compared with the recorded acoustic wave. Initial motion need not already have the steady periodic oscillation. ^[ref]

An elastic tongue fixed at one end and moving through a close-fitting opening without regularly striking its frame. An airflow can maintain oscillation by exchanging energy with the reed's elastic motion. This mechanism is an analogy for some aspects of phonation, not an anatomical identity.

Representation of data through sound. In an image encoder, frequency, amplitude, time and spatial auditory cues can label different image properties. Assigning several independent quantities to one auditory dimension can lose information unless the code supplies a way to separate them.

The conventional reverberation time $T_{60}$ is the time for the level of decaying acoustic energy to decrease by $60$ decibels, a factor $10^6$ in energy or $10^3$ in pressure wave amplitude. For exponential energy decay $E(t)=E(0)e^{-t/{\tau }_E}$, $T_{60}=6\ln (10){\tau }_E$. It is not the exponential relaxation time itself and cannot be inferred exactly from an unspecified audibility threshold.

Reverberation is the persistence of acoustic waves after a source stops, produced by repeated reflections in an enclosure. A diffuse-room approximation gives an approximately exponential decay of mean acoustic energy; an individual normal mode may instead be described as a damped resonator.

Acoustic attenuation is a reduction of a propagating sound signal beyond that attributed to the specified geometric spreading. Absorption converts acoustic energy to other forms, while scattering redirects it away from the receiving path. The inferred attenuation therefore depends on what the detector collects and how the spreading reference is defined.

For a plane wave with pressure amplitude $p(x)=p(0)e^{-\alpha x}$, the corresponding sound intensity decays as $I(x)=I(0)e^{-2\alpha x}$ in a fixed medium. An attenuation coefficient defined from intensity has twice the numerical value of this amplitude coefficient. Measuring the signal at several separations can estimate the slope, provided the source response, reflections and geometric corrections are controlled.

An ordinary acoustic wave involving a pressure and density oscillation in local thermodynamic equilibrium. In a Fermi liquid, this hydrodynamic response requires collisions fast compared with the oscillation period; zero sound instead describes a collisionless collective mode.

For a slowly varying parallel shear flow $U(y){\mathbf{e}}_x$, the leading WKB method neglects mean-velocity gradients while retaining local advection. The continuity equation and linearized Euler equations then give the displayed acoustic dispersion relation. On the positive intrinsic-frequency branch, $\omega =U{k}_x+c_0|\mathbf{k}|$ and the group velocity is $U{\mathbf{e}}_x+c_0\mathbf{k}/|\mathbf{k}|$. A wavefront normal is parallel to $\mathbf{k}$; the acoustic energy-ray direction generally differs because of the advection term.

In a stationary, horizontally homogeneous wall boundary layer, Hamiltonian ray-tracing equations conserve $\omega$ and $k_x$. If $U$ increases away from the wall, downstream-oriented $k_x>0$ waves have decreasing positive intrinsic frequency and decreasing $k_y^2$. At $U=\omega /k_x-c_0$, the transverse wavenumber vanishes and the acoustic ray turns. For $U(0)=0$, a downstream launch angle ${\beta }_0$ above the wall can reach a uniform stream of Mach number $M_{\infty }$ only if $\cos {\beta }_0\le (1+M_{\infty }{)}^{-1}$. This uses the positive acoustic branch, constant sound speed and no intervening critical layer; a turning point itself requires a uniform local approximation rather than the ordinary WKB method.

For a rigid rough boundary and incident pressure $A{e}^{i\alpha x-i\beta z}$, use time dependence $e^{-i\omega t}$ and ${\alpha }^2+{\beta }^2=k^2$. The flat reflected wave is $A{e}^{i\alpha x+i\beta z}$. Expanding the Neumann boundary condition at $z=0$ gives ${\partial }_z{\psi }_1=2A{e}^{i\alpha x}({\beta }^{2}h+i\alpha h^{\prime })$. For height transform $\hat{h}(q)=\int h(x)e^{-iqx}dx$, the outgoing correction is $(2A/(2\pi i))\int (k^2-\alpha q)\hat{h}(q-\alpha )e^{iqx+i\beta (q)z}/\beta (q)dq$, with $\beta (q)=\sqrt{k^2-q^2}$ on the outgoing sheet. Controlled height expansions also require adequate slope and spectral regularity; small incident phase displacement alone does not bound every normal-derivative correction.

For a zero-mean stationary rough height, the mean first-order correction vanishes. At second order, the boundary derivative is $-h{\partial }_z^2{\psi }_1+h^{\prime }{\partial }_x{\psi }_1$. Averaging this expression using the power spectrum of surface height gives the displayed coherent reflection coefficient. Thus the root mean square height alone does not fix the coherent mean: the height correlation spectrum matters. For roughness varying slowly enough that $q\simeq \alpha$ in the integral, $r_N\simeq 1-2{\beta }^2{\sigma }^2$. A local random-height phase average needs this additional approximation and is not a general consequence of stationarity.

Sonar uses sound propagation and reflected signals to detect objects or measure distances. Upward-looking sonar can record sea-ice draft along a moving platform's track; beam width, return selection and processing affect the inferred profile.

An acoustic waveguide confines sound transversely while allowing longitudinal propagation. Rigid walls impose a homogeneous normal-velocity condition, producing Neumann transverse modes. Each transverse eigenvalue gives a longitudinal wave number: modes below their cutoff decay evanescently, while cut-on modes carry outgoing sound. An open semi-infinite guide can be studied with the Wiener-Hopf method.

For the even acoustic waveguide mode $\cos (n\pi y/h)e^{-ikx}$, set $\sin \alpha =n\pi /(k_{0}h)$ and $k=k_0\cos \alpha >0$. At leading independent-edge order for large $k_{0}h$, the two incident travelling constituents each have amplitude $(-1{)}^n/2$ at the relevant wall endpoint. Reflecting the upper-edge coordinates reverses the edge-wave sign. Their path phases are $e^{\pm i{k}_{0}h\sin \theta }$. Thus Sommerfeld half-plane diffraction gives the displayed sum, where $D(r)=\sqrt{2/(\pi k_{0}r)}{e}^{-i{k}_{0}r-i\pi /4}$. For $n>0$, the apparent poles at $\theta =\pm \alpha$ are removable in the combined expression, with limit $ikhD(r)/2$. Single-edge nonuniform far-field poles are not physical infinities. This combined far-field limit assumes a radiation distance beyond the aperture's near field and neglects further edge interactions; it is not uniform at duct cutoff.

For rigid plates $x>0$, $y=\pm b$, an even incident plane wave travelling toward the opening can be scattered using Wiener-Hopf factorization. With time dependence $e^{i\omega t}$ and transform $\Phi (k)=\int \varphi (x)e^{ikx}dx$, take $\gamma =\sqrt{k^2-k_0^2}$ on the outgoing sheet. The common normal derivative on $y=b$ is supported on $x<0$; its transform is $D^{-}=i{L}^{+}(-k_0)L^{-}(k)/(k+k_0)$. The interior transform is $D^{-}\cosh (\gamma y)/(\gamma \sinh (\gamma b))$, while the exterior transform is $-D^{-}{e}^{-\gamma (|y|-b)}/\gamma$. These follow from continuity of normal velocity and the half-line pressure jump, followed by pole subtraction in a Wiener-Hopf equation and a finite-energy edge condition.

For fixed $|y|<b$ and $x\to -\infty$, the incident pole in the scattered transform cancels the analytic continuation of the incident plane wave. Near $k=-k_0$, the remaining transform is proportional to $(k+k_0{)}^{-1/2}$, so the total outgoing field has the displayed inverse-square-root cylindrical decay. A claim of square-root growth at negative infinity is incompatible with this outgoing solution.

For a separated acoustic potential with transverse Neumann eigenvalue ${\kappa }^2$, integration by parts gives $\int |{\nabla }_{\perp }F{|}^2={\kappa }^2\int F^2$. The time-averaged kinetic energy is proportional to $(k^2+{\kappa }^2)\int F^2$, while compression energy is proportional to ${\omega }^2\int F^2/c_0^2$. The dispersion relation ${\omega }^2=c_0^2(k^2+{\kappa }^2)$ proves equality. At a ray speed below $c_0$, nondegenerate stationary phase gives dispersive amplitudes of order $t^{-1/2}$; the zero transverse mode instead preserves an undispersed pulse on the sound-speed ray.

For an even field between rigid plates at $y=\pm b$, the transverse modes are $\cos (n\pi y/b)$, with longitudinal wave numbers $k_n=\sqrt{k_0^2-(n\pi /b{)}^2}$. With time dependence $e^{i\omega t}$, outgoing modes into $x>0$ use $e^{-i{k}_{n}x}$: choose positive real $k_n$ for cut-on modes and negative imaginary $k_n$ for cut-off modes. A lower-half-plane Fourier inversion extracts these modes by residues. Their decaying components are evanescent waves, not backward-propagating waves. At an exact cutoff one takes the radiation continuation before its limit.

An acoustic analogy rearranges fluid conservation equations into a chosen linear acoustic propagation operator plus effective forcing. The rearrangement can be exact before source approximations are made. A term classified as forcing can also contain propagation effects omitted from the reference operator; this produces propagation-source ambiguity in an acoustic analogy. The Lighthill acoustic analogy is a familiar homogeneous reference-medium example.

The distinction between generation and propagation depends partly on the reference operator chosen in an acoustic analogy. For example, a homogeneous-Laplacian operator with spatially varying reference sound speed omits the first-gradient terms of the stratified acoustic pressure equation. These terms then reappear as effective forcing, even for a freely propagating small disturbance in a static medium. A right-hand-side term is therefore not automatically an independent physical noise source.

Eliminating momentum between the continuity equation and conservation of momentum gives ${\rho }_{tt}={\partial }_i{\partial }_j(\rho u_i{u}_j-{\sigma }_{ij})+{\nabla }^2(p+\chi )$ when the force density is $-\nabla \chi$. For $q=p-(P-\chi )$ and time-independent reference fields,No energy equation is needed for this identity. The arbitrary reference fields should match the stationary medium where the outgoing sound is evaluated.

Small-height Dirichlet scattering expands the acoustic boundary condition about a mean plane for a fixed regular surface profile. The zeroth-order field includes the flat reflection; higher orders solve outgoing Helmholtz equation problems with induced boundary data. Small $|kh|$ controls the incident-field Taylor expansion, but rapidly varying roughness can require additional control of normal derivatives and spectral moments.

Through second order in height, a normally incident acoustic plane wave has the displayed mean reflected amplitude for a Dirichlet boundary. The propagating part of the power spectrum of surface height contributes a positive real correction to the negative flat reflection coefficient, while the evanescent part contributes an imaginary correction. The mean linear rough correction is zero.

The quadratic coherent correction weights the power spectrum of surface height by $\beta (q)$, so equal root mean square heights can give different mean fields. Only an additional long-spatial-scale approximation, with spectrum concentrated at $|q|\ll k$, reduces it to $2k^2{\sigma }^2$. Second-order surface scattering therefore contains information absent from first-order coherent reflection.

For a real zero-mean stationary height process, its power spectrum of surface height is the Fourier transform of the height covariance function. It is even and nonnegative, and ${\sigma }^2=(2\pi {)}^{-1}\int S_h(q)dq$. More general stationary processes have a spectral measure rather than a density. Weighted spectral moments control the existence of boundary derivatives and second-order mean scattering corrections.

For normal incidence on a Dirichlet rough surface, the scattered trace on the moving boundary is exactly $-e^{-ikh(x)}$. Its trace on the mean plane is instead $-1+2ikh+2kh\mathcal{B}h+\cdots$. The two traces are related by a normal Taylor expansion and must not be identified. Their ensemble averages can therefore have different quadratic coefficients without inconsistency.

For normal plane-wave incidence, the second-order Dirichlet boundary condition gives ${\psi }_s^{[2]}=-h{\partial }_z{\psi }_s^{[1]}$ at the mean plane, since ${\partial }_z^2{\psi }^{[0]}$ vanishes there. The Dirichlet-to-Neumann map for a Helmholtz half-space then yields $2kh\mathcal{B}h$. This nonlocal expression is not generally $2k^2{h}^2$; that local limit needs roughness varying slowly compared with the wavelength.

Taylor expansion of the Dirichlet boundary condition gives the first-order boundary trace $-h{\partial }_z{\psi }^{[0]}$ at the mean plane. The outgoing angular spectrum then propagates it. A zero-mean surface gives zero mean for this linear rough correction, while the complete scattered field still contains its nonzero flat reflection.

For incident amplitude one, ${\psi }_i=e^{i{p}_{i}x-i{q}_{i}z}$ with $q_i>0$, Taylor expansion of a small-height Dirichlet boundary condition gives ${\psi }_{\mathrm{sc}}^{[1]}(x,0)=2i{q}_{i}f(x)e^{i{p}_{i}x}$. If the mean height is constant, its mean transform is supported only at tangential wavenumber $p_i$, giving the displayed specular mean through first order. Small height alone is insufficient: deterministic mean height $a\cos Kx$ produces mean sidebands $p_i\pm K$. Stationary roughness has constant mean and therefore satisfies the first-order requirement.

First-order Dirichlet roughness has mean-plane trace $2i{q}_i{A}_{i}h(x)e^{i{p}_{i}x}$, where $p_i^2+q_i^2=k^2$ and $q_i>0$. Stationary phase applied to the outgoing angular spectrum evaluates the far-field amplitude at tangential wavenumber $k\sin {\theta }_s$. For a stationary surface illuminated over length $D$, the mean squared height transform divided by $D$ tends to its power spectrum of surface height. Thus the diffuse intensity samples the spectrum at the tangential momentum transfer. An infinite stationary surface has no finite total far-field amplitude without a window or an intensity-density convention.

The outgoing angular spectrum propagates prescribed scalar boundary data into a half-space. For $e^{-i\omega t}$ time dependence, $\beta (q)=\sqrt{k^2-q^2}$ has nonnegative real part on propagating modes and nonnegative imaginary part on evanescent modes. Its trace at the boundary is $g$, and each mode solves the Helmholtz equation. The spectral representation can be interpreted through distributions for nondecaying plane-wave data.

Use $\hat{\psi }(\nu )=\int \psi (x,0)e^{-i\nu x}dx$ and $q(\nu )=\sqrt{k^2-{\nu }^2}$ on the upward branch. If the uncentered field correlation is translationally homogeneous and $⟨\hat{\psi }(\nu )\hat{\psi }({\nu }^{\prime }{)}^{\ast }⟩=2\pi S(\nu )\delta (\nu -{\nu }^{\prime })$, its correlation at height $z$ is the displayed integral. The propagating spectrum is unchanged; evanescent contributions decrease as $e^{-2\sqrt{{\nu }^2-k^2}z}$. A spectral measure of a stationary random field permits discrete coherent components. Without correlation homogeneity a double spectral covariance integral depends on both observation coordinates rather than only their separation.

An upward angular spectrum obtained from a horizontal field trace is exact in the homogeneous half-space above that reference plane. For a rough boundary $z=f(x)$, choose the plane above $\sup f$. A trace at $z=0$ is not physically defined in the medium if any part of the surface lies above it. Using such a mean-plane trace requires an analytic continuation convention; convergence of the continued spectrum down to the rough boundary is an additional hypothesis. Small-height boundary perturbation uses the reference plane formally order by order and is distinct from an exact expansion throughout an arbitrary rough domain.

A progressive small-amplitude pressure perturbation in a uniform inviscid fluid depends on one travelling coordinate. Linear momentum balance gives its velocity parallel to propagation and proportional to pressure with impedance ${\rho }_0{c}_0$.

Recover a compactly supported acoustic source $Q$ from measurements of its outgoing field, such as its far-field pattern. The Helmholtz equation forward problem is specified with the Sommerfeld radiation condition. At a single frequency, the source-to-far-field operator at fixed frequency has invisible fixed-frequency nonradiating sources, and its compactness makes recovery of the visible component unstable.

A source is nonradiating at frequency $k$ if its outgoing field has zero far-field pattern. For every $\chi \in C_c^{\infty }(A)$, the source $Q=(\Delta +k^2)\chi$ has outgoing field $\psi =\chi$ and is invisible outside $A$. Integration by parts shows that its Fourier transform vanishes on $|\mathbf{\xi }|=k$, so it belongs to the kernel of a linear map defined by the source-to-far-field operator at fixed frequency. This family proves nonuniqueness independently of the instability of the inverse.

For $(\Delta +k^2)\psi =Q$, the outgoing Green function for the three-dimensional Helmholtz equation yields the far-field patternThe sign corresponds to the Green function satisfying $(\Delta +k^2)G=-\delta$. This Hilbert-Schmidt operator samples the source Fourier transform on the sphere $|\mathbf{\xi }|=k$; for a ball of radius $r_0$, its squared Hilbert-Schmidt norm is $r_0^3/3$.

The adjoint source-to-far-field operator composed with the source-to-far-field operator at fixed frequency has volume kernel $K(\mathbf{r},{\mathbf{r}}^{\prime })=(4\pi {)}^{-1}\mathrm{sinc}(k|\mathbf{r}-{\mathbf{r}}^{\prime }|)$, where $\mathrm{sinc}(s)=\sin (s)/s$ and $\mathrm{sinc}(0)=1$. This follows from ${\int }_{S_1}{e}^{ik\mathbf{\theta }\cdot \mathbf{d}}dS=4\pi \mathrm{sinc}(k|\mathbf{d}|)$. Its smoothing action helps explain the unstable normal equation for a linear inverse problem.

With complex inner products linear in the first argument, the adjoint operator of the source-to-far-field operator at fixed frequency isIt is a normalized Herglotz wave function. Complex conjugation reverses the phase sign, while retaining the real factor $-1/(4\pi )$.

An acoustic point source idealizes a compact emitter as a localized source of sound. The approximation is useful when its size is small relative to the relevant wavelength and observation distances; the actual emitter can be a vibrating object or a small driven volume of gas.

An isotropic outgoing wave in an unbounded lossless three-dimensional medium has pressure amplitude proportional to $1/r$ and sound intensity proportional to $1/r^2$ in the far field. Multiplying that intensity by a sphere's area gives a constant radiated power. The singular behavior at $r=0$ belongs to the idealization: the source's finite geometry and near field must replace it there.

An acoustic transducer converts between an acoustic disturbance and another form of signal or actuation. A microphone converts sound-induced motion into an electrical signal; a loudspeaker performs the opposite conversion. A piezoelectric transducer can use reciprocal coupling between deformation and an electric field to transmit and receive ultrasound.

Its frequency response, sensitivity and acoustic impedance affect what is emitted or measured. In a pulse-echo experiment, the same device can launch a short pulse and record the returning reflection. Travel time constrains distance only after the sound speed and the outgoing-plus-return path are accounted for.

A piezoelectric transducer converts mechanical stress into an electrical signal, or an applied electric field into mechanical deformation, using a piezoelectric material. The two directions enable both acoustic reception and transmission.

An ultrasound element is driven electrically to launch a pulse and then detects returning pressure waves through its generated charge or voltage. Mechanical resonance, electrical loading and acoustic coupling determine its bandwidth and sensitivity. The crystal response is therefore only one part of the complete instrument calibration.

A loudspeaker converts an electrical signal into motion that drives acoustic waves. In a moving-coil design, electric current through a coil in a magnetic field produces a force on the attached diaphragm. The diaphragm displaces air, creating sound pressure variations.

A small coil, magnet and lightweight cup can demonstrate the conversion directly. ^[ref] In an idealized uniform-field geometry, the driving force scales as $B\ell I$. The emitted sound also depends on diaphragm inertia, suspension spring constant, damping and acoustic impedance, so the electrical waveform alone does not specify a flat acoustic frequency response.

Time reversal acoustics uses recorded sound to send a field back toward the region that originally emitted it. An array of acoustic transducers records a pulse at multiple positions, reverses each recording in time, and emits the reversed signals from the corresponding positions. In a stationary reciprocal medium, the different paths can then contribute coherently near the original source.

A laboratory demonstration places a short-pulse source in a tank, records it at an array, removes or silences that source, and re-emits the reversed records. Mapping the returning pressure pulse tests the refocusing directly. Multiple propagation paths need not prevent focusing if they remain stable and are recorded; they can supply additional path diversity. Finite recording duration, limited bandwidth, incomplete spatial coverage and acoustic attenuation limit the reconstruction. Reversing recordings therefore implements a particular wave-control experiment, rather than undoing every dissipative process in the medium.

Multiple scattering can enlarge the angular diversity reaching a receiver array. Reversing the paths then makes the array act like a larger aperture, improving the focus in appropriate regimes of wave propagation in a random medium.

For a linear operator $H$ mapping transmitted signals to recorded signals, the positive operator $H^{\ast }H$ describes adjoint back-propagation after recording. In a reciprocal array, physical time reversal acoustics implements this operation up to a final complex conjugation and the chosen signal conventions.

The DORT method uses eigenvectors of a time reversal operator to focus on individual point scatterers. A separate nonzero eigenvalue corresponds to each scatterer in the ideal weak-scattering, well-resolved limit; coherent overlap and multiple scattering can change this correspondence.