Linear acoustics

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

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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 ${\gamma }^2=k^2-{\omega }^2/c_0^2$.

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.

For a massless spring-supported membrane adjoining a fluid half-space, kinematic and pressure balance give the dispersion relation $A{s}^4+s^2-1=0$ with $s=c/c_0$.

For $A{s}^4+s^2-1=0$, the physical speed approaches $c_0(1-A/2)$ as $A$ tends to zero and scales as $c_0{A}^{-1/4}$ as $A$ tends to infinity.

A disturbance decaying over depth $1/k$ accelerates fluid mass of order ${\rho }_0/k$ per unit area, giving a frequency-dependent added inertia.

For real harmonic fields with complex amplitudes $\hat{p}$ and $\hat{u}$, their period-averaged product is $\mathrm{Re}(\hat{p}{\hat{u}}^{\ast })/2$.

When pressure and velocity are in quadrature, their instantaneous energy flux oscillates but its period average vanishes.