Acoustics

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.

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 $-1/(4\pi )$.

A loudspeaker converts an electrical signal into sound. It can drive an acoustic wave whose frequency is controlled independently of the spatial measurement.

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.