Linear acoustics

Linear acoustics describes small pressure, density, and velocity perturbations about a uniform quiescent compressible fluid.

A velocity potential obeying the wave equation with constant speed of sound $c_0$ has spherical normal modes $\varphi =e^{-i\omega t}[A\cos kr+B\sin kr]/r$, $k=\omega /c_0$. Zero radial velocity at radii $a,b$ gives Neumann conditions on the potential, not on its numerator. The inner condition is built into $F(r)=C[\cos k(r-a)+(ka{)}^{-1}\sin k(r-a)]/r$. The outer condition gives $k(b-a)\cos k(b-a)-(1+k^{2}ab)\sin k(b-a)=0$, equivalent to the displayed equation for nonzero oscillatory modes. The formal zero root is a constant potential gauge, rather than a nonzero-frequency acoustic wave. Applying a one-dimensional sine condition to the numerator would miss the spherical geometric factors.

A wall displacement $\eta$ with spring law $p^{\prime }={\rho }_{0}k\eta$, combined with linear Bernoulli $p^{\prime }=-{\rho }_0{\varphi }_t$ and kinematic condition ${\varphi }_n={\eta }_t$, gives the displayed dynamic boundary condition. For a rectangular cavity of width $a$, height $b$, rigid remaining walls and sound speed $c_0$, real vertical modes obey ${\omega }^2=c_0^2[(n\pi /a{)}^2+(\beta /b{)}^2]$ and $\beta \tan \beta =-{\omega }^{2}b/k$. On each interval $((j+1/2)\pi ,(j+1)\pi )$ the tangent equation has a real root. As $c_0^2/(kb)\to \infty$ at fixed mode numbers, these branches tend to $\beta =(j+1/2)\pi$, the pressure-release upper-wall acoustic spectrum. Imaginary vertical exponents can describe additional branches and are not counted by this real-root argument.

For a uniform background velocity $U_0$ and speed of sound $c_0$, linearized conservation of mass, the Euler equation and the local equation of state $p_1=c_0^2{\rho }_1$ give this wave equation. In the coordinate $x-U_{0}t$ it is the ordinary one-dimensional wave equation. Its two characteristic curves travel at laboratory velocities $U_0+c_0$ and $U_0-c_0$; both point downstream in supersonic flow.

Sound pressure is the deviation of pressure from the undisturbed background. In linear acoustics, it is small compared with the background pressure; a microphone measures a response to this varying pressure, rather than directly recording fluid displacement.

The logarithmic level of root-mean-square sound pressure relative to a reference, conventionally $20\mu \mathrm{Pa}$ in air. A pressure ratio $r$ changes level by $20{\log }_{10}r$ decibels. The factor twenty follows from the square dependence of acoustic intensity on pressure in a fixed medium, combined with the power-level convention $10{\log }_{10}$.

The local fluid velocity perturbation in an acoustic wave. It describes the oscillating material, whereas the speed of sound describes propagation of the disturbance. A fluid parcel need not travel one wavelength during a wave period.

The small displacement of a fluid parcel from its undisturbed position in a linear acoustic model. Its time derivative is the acoustic particle velocity.

A Kundt's tube turns an invisible standing wave into a measurable spatial pattern. A driven air column supports a longitudinal wave; measuring its frequency and the spacing of repeated nodes of a standing wave gives the speed of sound. The same drive can compare the wavelengths in a gas and a vibrating solid. August Kundt's 1866 paper introduced acoustic dust figures; its footnote explicitly counts successive nodes of a standing wave as half a wavelength.^[ref]

The attractive feature is the separation of measurements: a clock or frequency counter measures repetition in time, while a ruler measures repetition in space. Their product supplies a propagation speed. The pressure pattern and the fluid-motion pattern are different, however; identifying which one a detector responds to prevents a quarter-wavelength mistake.

PHYWE's implementation rubs a $1.60\mathrm{m}$ rod supported $0.40\mathrm{m}$ from each end. Its end disc drives a dusty glass tube; a movable tuning slide selects a clear pattern. A ruler measures repeated positions and a thermometer records gas temperature. The two supports select a rod wavelength equal to its length.^[ref]

The University of Iowa also documents an electronic version: a loudspeaker drives the tube and a microphone on a plunger supplies an oscilloscope readout. This permits a spatial scan of sound pressure without relying on the dust pattern.^[ref]

For a proposed pressure-scan measurement, keep the drive frequency fixed and move a small microphone through the tube. Record its position and fitted sinusoidal amplitude after each move; repeat the scan in reverse to reveal drift. Locate several consecutive maxima, rather than estimating a wavelength from a single pair. Reducing the drive checks whether the inferred spacing is independent of amplitude in the linear acoustics regime. A drive-off record estimates electrical background. These are controls for the proposed measurement, not reported observations from the cited demonstrations.

A microphone and its access opening can perturb the field. End compliance, losses and transverse modes also challenge the one-dimensional description. Dust traces identify a repeating structure, but the grains are not fluid parcels: the equations below describe sound pressure and acoustic particle velocity, rather than a detailed grain-transport model. The temperature and gas composition must remain sufficiently uniform to interpret a single fitted speed of sound.

Consider a uniform tube $0\le x\le L$ with rigid ends, constant undisturbed mass density ${\rho }_0$ and speed of sound $c$. Neglect losses and transverse structure. A free normal mode hasand a consistent choice of sound pressure, acoustic particle velocity and acoustic displacement isHere $P>0$ is a peak sound pressure. Direct differentiation verifies ${\rho }_0{\partial }_{t}u=-{\partial }_x{p}^{\prime }$ and ${\partial }_t{p}^{\prime }=-{\rho }_0{c}^2{\partial }_{x}u$, the one-dimensional linear homentropic acoustic equations. The rigid ends enforce $u=0$ there. A driven experiment only approximates this free-mode geometry; the formulas do not predict a finite resonant amplitude for an undamped driven tube.

The maxima of sound pressure amplitude occur at $kx=j\pi$, where acoustic particle velocity vanishes. The maxima of acoustic particle velocity amplitude instead occur at $kx=(j+1/2)\pi$, where sound pressure vanishes. The two families are displaced by $\lambda /4$, but each family repeats after $\lambda /2$. At a position where both are nonzero, the time dependence is also a quarter cycle apart. Using consecutive maxima of sound pressure amplitude therefore givesThe relation also applies to consecutive minima of sound pressure amplitude, provided the same family is used throughout. Mixing maxima and minima would halve the spacing and give an erroneous speed of sound.

The local acoustic energy density is $p^{\prime 2}/(2{\rho }_0{c}^2)+{\rho }_0{u}^2/2$. Averaging over one wave period gives $P^2/(4{\rho }_0{c}^2)$, independent of $x$. Likewise $⟨p^{\prime }u⟩=0$: this ideal standing wave has no mean longitudinal acoustic energy flux, although its pressure and fluid velocity oscillate. Energy transport into a real lossy tube is a different question from the free-mode pattern.

Write the measured positions of consecutive maxima of sound pressure amplitude as $x_j=a+jd+{\epsilon }_j$. The arbitrary origin $a$ is fitted along with the half-wavelength $d$ by linear regression; forcing the line through zero would turn a ruler-origin offset into an erroneous slope. For independent equal-variance position errors of standard deviation ${\sigma }_x$,As an original example, choose $f=1000\mathrm{Hz}$, $a=24\mathrm{mm}$ and $d=171.5\mathrm{mm}$. Add stipulated position errors $(1,-1,0,0,-1,1)\mathrm{mm}$ at $j=0,\dots ,5$. Their sum and index-weighted sum vanish, so the fitted values remain $\hat{a}=24\mathrm{mm}$ and $\hat{d}=171.5\mathrm{mm}$, giving $c=343.0m/s$. These are synthetic inputs, not measurements of a particular air sample.

With stipulated ${\sigma }_x=1\mathrm{mm}$ and independent drive-frequency uncertainty ${\sigma }_f=1\mathrm{Hz}$, first-order uncertainty propagation givesThis statistical calculation omits common scale-calibration error, thermal drift and mode misidentification. Counting more positions reduces independent localization error; it cannot correct a wrong family of extrema or an incorrectly calibrated ruler.

A freely moving rod end has zero axial stress, not zero axial displacement. For a uniform slender rod of length $\ell$, its low-frequency extensional modes have shapes ${\xi }_r(x)\propto \cos (n\pi x/\ell )$ and wavelengths ${\lambda }_r=2\ell /n$. A centre clamp is compatible with the $n=1$ displacement node and ${\lambda }_r=2\ell$; clamps at $\ell /4$ and $3\ell /4$ are compatible with $n=2$ and ${\lambda }_r=\ell$. The support positions therefore matter before a gas wavelength can determine a rod speed of sound.

For a chosen model with $E=200\mathrm{GPa}$, ${\rho }_r=8000kg/{\mathrm{m}}^3$ and $\ell =1.60\mathrm{m}$, the slender-rod extensional relation $c_r=\sqrt{E/{\rho }_r}$ gives $5000m/s$. The two-quarter-support mode then has $f=3125\mathrm{Hz}$. If the gas has stipulated $c_g=343m/s$, its half-wavelength is $d_g=54.88\mathrm{mm}$, andThe centre-supported fundamental would instead have $f=1562.5\mathrm{Hz}$ and twice that gas spacing. The same rod material has not changed its wave speed. In this slender-rod model, $E$ is Young's modulus; a bulk compressional wave involves a different elastic-modulus combination, so the two wave speeds are not interchangeable.

Linearizing inviscid barotropic flow about uniform rest gives ${\partial }_t(\nabla \times u)=0$, not automatically $\nabla \times u=0$. Irrotational linear acoustics therefore requires zero initial vorticity or restriction to the longitudinal acoustic mode. A time-independent divergence-free field, with zero pressure and density perturbations, is a nonacoustic solution of the linearized equations.

An aperture in a rigid plane with prescribed harmonic normal displacement $f(x^{\prime })e^{i\omega t}$ radiates into a half-space. Superposing acoustic monopoles, each doubled by its same-sign rigid-wall image, gives the stated integral, where $R$ is the distance from aperture element to observer. In the acoustic far field, expanding $R$ to first order across the aperture identifies the angular radiation amplitude with the aperture's Fourier transform.

The acoustic far field is the radiation region where the observer is many reduced wavelengths from the source and far compared with its geometric extent. Three-dimensional outgoing acoustic density perturbations spread as $R^{-1}$; two-dimensional harmonic fields spread as $R^{-1/2}$. The condition is independent of the acoustic compact-source approximation $k_0\ell \ll 1$. Differentiating a retarded phase produces the radiating part of a Green function integral; derivatives of the spreading factor produce near-field terms.

Acoustic directivity describes the angular dependence of radiation in the acoustic far field. For a time-harmonic source in $d$ spatial dimensions, separating the geometric spreading and oscillatory phase leaves a complex amplitude pattern $\mathcal{D}$. The angular intensity pattern is proportional to $|\mathcal{D}{|}^2$. Directivity can be normalized by a reference direction or total radiated power; a stated convention is needed when comparing absolute directivity factors.

For a source surface parameterized by $y(p,q,\tau )$, the retarded time equation is $t=\tau +|x-y(p,q,\tau )|/c_0$. Integrating a Dirac delta distribution enforcing that equation produces the reciprocal derivative $|1-M_r{|}^{-1}$ at each simple root. For subsonic motion there is a unique retarded root. Multiple roots require summation; a zero derivative is outside the simple-root rule. This factor is the moving-source counterpart of a delta change-of-variable Jacobian.

For adiabatic perturbations of a static perfect gas, pressure obeysThe stationary balance is $\nabla (p_0+\chi )=0$. Density and entropy need not be spatially uniform. Expanding the divergence exposes the gradient terms that distinguish this equation from the homogeneous acoustic pressure wave equation. Its divergence form yields weighted acoustic Green-function reciprocity.

An acoustic density perturbation is the deviation of fluid mass density from its quiescent reference value. In a homogeneous isentropic linear acoustic field, its corresponding pressure perturbation is $p^{\prime }=c_0^2{\rho }^{\prime }$. Nonlinear and entropy-dependent departures from this relation enter the Lighthill stress tensor.

An acoustic source is compact when its size is much smaller than the acoustic wavelength. The propagation delay across the source can then be neglected in leading source integrals. This condition is distinct from the radiation-region requirement $k_{0}r\gg 1$.

Acoustic sources may be organized by the number of spatial derivatives in their effective wave equation forcing. A direct scalar source is an acoustic monopole, a divergence of a force density an acoustic dipole, and a double divergence of a stress tensor an acoustic quadrupole.

An acoustic quadrupole has forcing ${\partial }_i{\partial }_j{T}_{ij}$. The acoustic compact-source approximation reduces its leading far-field density to $n_i{n}_j{\ddot{S}}_{ij}/(4\pi c_0^{4}r)$ in three dimensions, with $S_{ij}=\int T_{ij}{d}^{3}y$ evaluated at retarded time.

Two diametrically opposite rotating blades with equal tangential loads have zero total rotating force. Their leading compact acoustic dipoles cancel, but their retarded positions and moving-surface retarded Jacobians leave an acoustic quadrupole. For line load $2Fr/a^2$ and angular speed $\Omega$, the first radiating density is $2F{\Omega }^{2}a{\sin }^2\theta \sin (2\Omega {\tau }_0)/(3\pi c_0^{4}R)$. The twofold geometry gives frequency $2\Omega$, while projection into the rotor plane gives the amplitude directivity ${\sin }^2\theta$.

For a low-Mach number source with stress $O({\rho }_0{U}^2)$ and advective time $\ell /U$, the three-dimensional compact acoustic quadrupole has density amplitude $O({\rho }_0{M}^4\ell /r)$, where $M=U/c_0$. The power depends on these source-time and stress hypotheses, not only on the quadrupole label.

In two dimensions, the outgoing harmonic Helmholtz equation kernel has magnitude proportional to $(k_{0}r{)}^{-1/2}$. Combining this spreading with the compact acoustic quadrupole source and $k_0\ell =O(M)$ changes the density-amplitude Mach number power to $7/2$ when geometric range is separated.

An acoustic dipole represents an applied force density. A surface force defect contributes $-{\partial }_i(L_i{\delta }_s)$ to the density wave equation.

For a small body of size $l$ in flow of speed $u$, force of order ${\rho }_0{u}^2{l}^2$ varying on time $l/u$ produces an acoustic dipole density amplitude of the displayed order, where $M=u/c_0$ is the Mach number. Its sound power scales as ${\rho }_0{u}^6{l}^2/c_0^3$. The corresponding compact acoustic quadrupole Mach-number scaling has density amplitude of order $(l/r)M^4$ and power ${\rho }_0{u}^8{l}^2/c_0^5$.

An acoustic monopole represents unsteady mass or volume injection. A moving-interface mass-flux defect contributes ${\partial }_t(Q{\delta }_s)$ to the density wave equation.

A point source moving at angular speed $\Omega$ on a circle of radius $a$ has observer-direction Mach number $m=\Omega a\sin \iota /c_0$. In the far field its retarded phase satisfies $\Theta +\theta -\pi /2=\Omega (t-r/c_0)+m\sin \Theta$. For $m<1$ there is one retarded root, and a constant source strength gives ${\rho }^{\prime }=q\dot{\Theta }/(4\pi c_0^2\Omega r)$. For $m>1$, multiple retarded roots can occur; their contributions require $|1-m\cos \Theta |$ and a sum over roots. Coalescing roots produce a caustic in the ideal point-source model.

The Lighthill acoustic analogy rewrites fluid mass and momentum conservation as a constant-speed wave equation driven by a nonlinear Lighthill stress tensor. It is an exact rearrangement before making source approximations. The outgoing field can be constructed with a retarded acoustic Green function.

If the continuity equation has mass source $M$ and the conservative momentum equation has momentum source $S_i$, elimination of momentum gives the displayed wave equation. With no imposed momentum source, its extra term is ${\partial }_{t}M$. A compact source of total mass rate $\mathcal{M}=\int MdV$ radiates the acoustic monopole ${\rho }_M^{\prime }=\dot{\mathcal{M}}(t-r/c_0)/(4\pi c_0^{2}r)$. The momentum source supplies an additional acoustic dipole; specifying mass injection alone does not fix that dipole.

For compact Lighthill stress tensor and applied-force distributions, define $S_{ij}=\int T_{ij}{d}^{3}y$ and $F_i=\int f_i{d}^{3}y$. Differentiating the retarded acoustic Green function at far distance gives the displayed leading radiation field, evaluated at $t-r/c_0$. Source compactness requires its size to be small compared with the acoustic wavelength, not merely with the observer distance.

Mass and momentum flux defects at a moving interface add surface acoustic monopole and acoustic dipole terms to Lighthill acoustic analogy. For a physical conservative fluid shock wave, the Rankine-Hugoniot conditions set both defects to zero; the shock still affects distributional derivatives of the Lighthill stress tensor. A discontinuity alone is not independent mass or force injection.

The Lighthill acoustic analogy for a moving body has a volume acoustic quadrupole, a surface acoustic monopole from acoustic thickness noise, and a surface acoustic dipole from acoustic loading noise. For an impermeable moving surface, with outward normal $n$, normal surface speed $v_n$, and fluid and body normal velocities equal, the surface mass coefficient is ${\rho }_0{v}_n$ and the surface loading is the force exerted on the fluid. Schematically, the density-source equation isThe density perturbation is understood with the chosen interior extension. Permeable-surface versions have additional mass and momentum flux terms. Source approximations must distinguish local thickness radiation from cancellation of its compact net-volume contribution.

For a fixed impermeable object, the Ffowcs Williams-Hawkings equation has a volume acoustic quadrupole and a surface acoustic dipole, with no mass-injection term. If $F$ is the total force exerted on the fluid and $Q=\int TdV$ the integrated Lighthill stress tensor, the acoustic compact-source approximation in the acoustic far field gives the displayed pressure at retarded time. A force convention directed onto the body reverses the loading sign. The retarded acoustic Green function carries $1/c_0^2$ when the wave operator is ${\partial }_t^2-c_0^2\Delta$.

Loading noise is the acoustic dipole radiation of the force exerted by a body on its surrounding fluid. For a compact, slowly moving source, the radiating acoustic density perturbation is $n\cdot \dot{\mathcal{F}}(t-R/c_0)/(4\pi c_0^{3}R)$, where $\mathcal{F}$ is the total force. If this total is constant, finite source-delay corrections can still create higher acoustic multipole sources.

For a nonseparating small body in a smooth nearly inviscid eddy of speed $U$ and length $\ell$, the fluctuating resultant is an added mass or pressure-gradient force $F\sim {\rho }_0{a}^3{U}^2/\ell$. A uniform pressure integrates to zero over a closed surface. With time $\ell /U$, its acoustic dipole power divided by the eddy's acoustic quadrupole power is $(a/\ell {)}^6/M^2$, where $M=U/c_0$ is the Mach number. Equal powers therefore give the displayed threshold. An area-scaled separated drag force with the same eddy time instead gives $a/\ell \sim \sqrt{M}$; a body-scale time gives a different criterion. The threshold depends on the specified loading and time models.

Thickness noise is the sound generated by displacement of fluid by a moving impermeable body, represented by the surface acoustic monopole term. For a rigid body, ${\int }_S{v}_{n}dS=d\mathcal{V}/dt=0$, so its leading acoustic compact-source approximation has no net-volume monopole. Its local source need not vanish: a moving finite-volume body can radiate through higher spatial moments. A thin-body or negligible-volume approximation is needed when all such radiation is to be omitted.

In the incompressible near field of a spherical bubble, $u_r=a^2\dot{a}/s^2$. Its volume-flux acoustic monopole is the displayed term at retarded time. The integrated surface pressure force vanishes by spherical symmetry, while the Lighthill stress tensor approximation ${\rho }_0{u}_i{u}_j$ has moment $S_{ij}=4\pi {\rho }_0{a}^3{\dot{a}}^2{\delta }_{ij}/3$. Its leading compact acoustic quadrupole contribution is ${\rho }_0(a^3{\dot{a}}^2{)}^{\prime \prime }/(3c_0^{4}r)$. For small fractional radius oscillation $\delta$ and compactness parameter $\alpha =\omega a_0/c_0$, its size relative to the leading monopole is $O(\delta {\alpha }^2)$.

For a small rigid sphere translating with velocity $\mathbf{V}(t)$, the integrated thickness source ${\rho }_0\int \mathbf{V}\cdot {\mathbf{n}}_{s}dS$ vanishes. The first spatial moment is instead ${\rho }_0\int y_j\mathbf{V}\cdot {\mathbf{n}}_{s}dS={\rho }_0\mathcal{V}{V}_j$, where $\mathcal{V}$ is the sphere volume. Expanding the retarded time across the compact source gives the displayed first nonzero thickness field. It is an acoustic dipole, at the same order as acoustic loading noise for a translating sphere. The added mass of a sphere supplies half this volume-mass coefficient through the loading term; their sum is $3{\rho }_0\mathcal{V}\mathbf{n}\cdot \ddot{\mathbf{V}}/(8\pi c_0^{3}r)$.

This tensor contains convective momentum flux, departure from the reference linear pressure-density relation, and viscous stress. Its double divergence is an acoustic quadrupole distribution in Lighthill acoustic analogy.

Write a piecewise smooth Lighthill stress tensor as $T_{ij}=T_{ij}^{-}+D_{ij}H(S)$, where $D_{ij}=T_{ij}^{+}-T_{ij}^{-}$ and the regular level surface $S=0$ is a shock wave. The distributional derivative of the Heaviside step function givesHere repeated indices are summed, $S_i={\partial }_{i}S$ and $S_{ij}={\partial }_i{\partial }_{j}S$. Keep the coefficients as smooth extensions before multiplying distributions: prematurely replacing the coefficient of ${\delta }^{\prime }(S)$ by its surface trace can lose a $\delta (S)$ contribution. The extra surface delta distributions and their derivatives are supported entirely on the discontinuity.

For perturbations $({\rho }^{\prime },p^{\prime },\mathbf{u})$ about a uniform fluid of density ${\rho }_0$, the linearized continuity equation, momentum equation, and homentropic pressure perturbation relation areFor irrotational flow $\mathbf{u}=\nabla \varphi$, they imply $p^{\prime }=-{\rho }_0{\varphi }_t$ and ${\varphi }_{tt}=c_0^2{\nabla }^2\varphi$.

Eliminating the velocity and density perturbations from the linear homentropic acoustic equations gives

For harmonic convention $e^{i\omega t-ikx}$, choose $\mathrm{Re}\gamma >0$ by continuation from $\mathrm{Im}\omega <0$ and impose vertical factors $e^{-\gamma y}$ above the interface and $e^{\gamma y}$ below it. At positive real frequency, $\gamma =i\sqrt{k_0^2-k^2}$ for propagating Fourier components and is positive real for evanescent components. This analytic prescription fixes the radiation branch and how the Fourier contour passes poles.

For a homentropic reference state, first-order pressure and density perturbations satisfy $p^{\prime }=c_0^2{\rho }^{\prime }$, where $c_0$ is the sound speed.

For irrotational linear acoustic flow, $\mathbf{u}=\nabla \varphi$, $p^{\prime }=-{\rho }_0{\varphi }_t$, and $\varphi$ obeys the wave equation.

For $R(t)=a+\mathrm{Re}(ϵe^{i\omega t})$, put $k=\omega /c_0$. To first order in $ϵ/a$, the exterior outgoing velocity-potential amplitude isIt satisfies ${\hat{u}}_r(a)={\hat{\varphi }}^{\prime }(a)=i\omega ϵ$.

The mean power radiated by the pulsating sphere is

At $r=a$,For $\omega a/c_0\gg 1$, kinetic and compressional energies are equal and the field is locally radiative. For $\omega a/c_0\ll 1$, reactive kinetic energy dominates and the mean radiated flux is small.

For $R<r<2R$, a rigid inner sphere and harmonic pressureproduce a standing radial field. With $k=\omega /c_0$ and $\alpha =kR$,The pressure and radial velocity are in temporal quadrature, so the period-averaged acoustic intensity vanishes.

The ideal pressure-forced response becomes resonant whenequivalently $\tan \alpha =-\alpha$. Damping regularizes the divergent linear response and introduces a nonzero phase lag and mean supplied power.

Let a thin spherical shell have equilibrium radius $a_0$, mass $m$ per unit area, and radial restoring stiffness $\kappa$ per unit area. If it encloses gas of density ${\rho }_0$ and sound speed $c_0$, its spherically symmetric normal-mode frequencies obeyThe fraction is the frequency-dependent acoustic added mass of the enclosed gas.

Linear acoustic fields obeywhere

For outgoing velocity potential $\varphi =F(r-ct)/r$, the radial velocity is $F^{\prime }/r-F/r^2$ and the pressure perturbation is ${\rho }_{0}c{F}^{\prime }/r$. The acoustic energy density separates into kinetic density ${\rho }_0(F^{\prime }/r-F/r^2{)}^2/2$ and compressional density ${\rho }_0(F^{\prime }/r{)}^2/2$. They are generally unequal near the source but agree at leading far-field order. The additional $r^{-2}$ velocity term is the spherical near field.

For a linear acoustic perturbation of a uniform fluid,is the sum of kinetic and compressional energy densities. Together with the acoustic energy flux $\mathbf{I}=p^{\prime }\mathbf{u}$, it satisfies $E_t+\nabla \cdot \mathbf{I}=0$.

A surface wave with phase speed below the bulk sound speed decays normally as $e^{-\alpha z}$, where $\alpha =k\sqrt{1-c^2/c_0^2}$.

A membrane of mass per area $m$ and tension $T$ separating two identical acoustic half-spaces has evanescent modes withThe second term is the added mass of an evanescent fluid layer contributed by both sides. Hence the phase speed is below the vacuum membrane speed $\sqrt{T/m}$, and for $k\to 0$ it obeys $c\sim \sqrt{Tk/(2{\rho }_0)}\to 0$. Pressure and normal velocity are in quadrature, so the mean normal acoustic energy flux vanishes.

The in-plane tensile force per unit transverse length in a thin elastic sheet produces the linear transverse restoring force $T{\eta }_{xx}$ per unit area. A spatially uniform displacement has zero curvature and hence no elastic-sheet tension force. This explains why infinite elastic-sheet tension blocks fixed nonzero-wavenumber acoustic membrane waves but does not make the uniform mode rigid. The sheet's inertia, by contrast, acts also on uniform oscillations.

A harmonic line force localized at one point of a two-dimensional sheet gives $\hat{\eta }=F/D$ and $\Delta =\gamma D$ in the Fourier transform. The outgoing upper pressure is the displayed amplitude. Dividing by $c_0^2$ converts it to acoustic density perturbation. The method of steepest descent gives cylindrical sound from the saddle point, while crossed zeros of the dispersion relation give guided or leaky modal residues. Retaining $k_0=\omega /c_0$ is essential for dimensionally consistent spreading and prefactors.

For one acoustic half-space and vacuum on the other side, a harmonic point force gives displacement amplitude $F\gamma /D$ and fluid acoustic velocity potential amplitude $i\omega F{e}^{-\gamma y}/D$. For time dependence $e^{-i\omega t}$, the outgoing acoustic square-root branch has $\gamma >0$ for $|k|>k_0$ and $\gamma =-i\sqrt{k_0^2-k^2}$ for $|k|<k_0$. The causal limit $\omega \mapsto \omega +i0$ places the positive guided-mode pole above the inversion contour and the negative pole below. The saddle point generates cylindrical radiation, whereas guided residues decay normally to the membrane.

A sheet of mass per area $m$ and elastic-sheet tension $T$ backed by an identical outgoing fluid half-space presents the stated surface acoustic impedance to the other half-space, using upward sheet velocity $i\omega \eta =-V$. The fluid term is its normal acoustic impedance; the remaining term is structural. Infinite elastic-sheet tension suppresses every fixed nonzero spatial wavenumber, but not $k=0$, whereas infinite mass suppresses even spatially uniform motion. A massless untensioned sheet between identical fluids is transparent.

A free acoustic membrane wave is a zero of the outgoing dispersion relation $D$. At this zero, the impedance seen from one side equals the negative of that side's outgoing normal acoustic impedance. Thus the continued by analytic continuation reflection coefficient has a pole. A real guided mode is evanescent normal to the sheet; a complex root can describe a leaky mode. The pole describes nontrivial homogeneous motion with no incoming wave, rather than unbounded passive reflection at an ordinary real incidence angle.

A pinned semi-infinite membrane scatters an incoming guided mode into reflected guided modes and outgoing sound. For unit incident displacement the scattered endpoint value is $-1$. Its Half-range Fourier transform therefore has nonzero endpoint terms even though the total displacement vanishes. The Wiener-Hopf equation kernel is $2/(m{\omega }^2-T{k}^2)+\gamma /({\rho }_0{\omega }^2)$ with the outgoing branch ${\gamma }^2=k^2-{\omega }^2/c_0^2$.

For identical unit-density, unit-sound-speed fluids above and below a sheet on $x<0$, use time $e^{i\omega t}$ and transform $\int e^{ikx}\varphi dx$. The outgoing acoustic square-root branch obeys ${\gamma }^2=k^2-{\omega }^2$. If $J$ is the pressure jump upper minus lower, $s={\eta }^{\prime }(0)$ and $d^{+}$ the right Half-range Fourier transform of the scattered normal derivative, the Wiener-Hopf equation isThe total displacement is pinned, so its transformed second derivative is $s-k^2{\eta }^{-}$. The bare-sheet pole $b=\omega \sqrt{m/T}$ in reconstructed displacement must cancel: $J(b)=Ts$. This is an incoming-pole cancellation at a pinned membrane edge condition, distinct from the outgoing coupled-mode pole at the zero of $K$.

Reconstructing the left scattered displacement in fluid-loaded membrane edge scattering divides by $m{\omega }^2-T{k}^2$. The pole at the lower bare-membrane wavenumber $b=\omega \sqrt{m/T}$ must cancel to preserve outgoing half-plane analyticity. This gives a linear condition determining the otherwise unknown endpoint slope $A$. The prescribed incident pole in the total field is a different singularity.