Acoustic wave

An acoustic wave is a propagating disturbance in a material's pressure, mass density and particle motion. Compressing a small fluid region raises its pressure; the resulting pressure gradient accelerates adjacent material, while inertia carries the motion onward. This restoring mechanism permits longitudinal waves in an ordinary fluid.

For small perturbations about a uniform resting state, the linear continuity equation and momentum conservation combine into a wave equation. The squared speed of sound is the relevant derivative of pressure with respect to mass density; rapid locally reversible compression usually uses the isentropic derivative. A microphone records the pressure variation, while a travel-time measurement probes propagation speed. The particle velocity can remain much smaller than the speed of sound: local material oscillation and movement of the wave pattern are different quantities. Acoustic energy flux relates the pressure and particle-velocity fields to transported power.

For a rigid cylinder of radius $a$ and height $h$, the velocity potential obeys the wave equation with Neumann boundary conditions. Separation in cylindrical coordinates gives radial factors $J_m(k_{m,n}r/a)$, angular sine and cosine factors of integer order $m$, and axial factors $\cos (\ell \pi z/h)$. The nonzero radial numbers satisfy $J_m^{\prime }(k_{m,n})=0$, yielding the displayed frequencies.

There is also a constant radial mode, with $m=0$ and radial number zero. A formal zero of $J_m^{\prime }$ at the origin for higher $m$ must not be counted by substituting $k=0$ into $J_m(kr)$, which would give the zero function. Solving the zero-radial-eigenvalue equation directly leaves only the constant mode under the side-wall condition. The fully constant potential has zero velocity and is a gauge mode; constant radial modes with nonzero axial index are physical longitudinal oscillations.

Nonlinear acoustics retains finite-amplitude effects in sound waves, including amplitude-dependent propagation speed, distortion and possible shock waves. For a one-dimensional polytropic flow with $p\propto {\rho }^{\gamma }$, a rightgoing simple wave has characteristic speed $u+c=c_0+(\gamma +1)u/2$ relative to a quiescent reference. Thus small-amplitude linear acoustics need not be uniform when a background velocity nearly cancels the speed of sound.

Ultrasound is sound above the conventional upper limit of human hearing, approximately $20\mathrm{kHz}$. In a crystal, ultrasonic travel-time measurements determine long-wavelength elastic wave speeds, which are the small-wavevector slopes of the corresponding acoustic phonon branches.

Acoustic measurement of a moving-scatterer velocity projection using frequency or phase changes. A speed estimate needs the scattering geometry; finite pulsed sampling can alias the measured shift.

An emitted acoustic pulse is followed by detection of returning echoes. Travel time estimates depth under a declared sound-speed and path model, while echo amplitudes depend on interfaces and propagation losses.

An M-mode ultrasound scan repeatedly samples one pulse-echo ultrasound beam and places successive depth profiles alongside one another in time. Depth is one display axis and acquisition time the other; echo strength controls brightness. This makes motion along the selected line visible. ^[ref]

A stationary interface traces a line at constant depth; an oscillating interface traces a varying depth. Converting echo delay to depth still assumes a sound speed and a return path. This display records one beam over time, whereas a B-mode ultrasound scan combines beam lines across space.

A B-mode ultrasound scan maps echo strength to displayed brightness and assembles multiple pulse-echo ultrasound lines into a spatial image. Each echo's location is inferred from beam direction and return time using an assumed sound speed. ^[ref]

A bright point represents a strong received echo, not a direct measurement of tissue density. Acoustic impedance contrasts, propagation losses and the instrument's display mapping all affect brightness. A moving image consists of repeated spatial frames; in an M-mode ultrasound scan, the second display axis instead tracks time along one beam.

An A-mode ultrasound scan plots the returning echo amplitude along one pulse-echo ultrasound beam. With assumed sound speed $c$, echo delay $\tau$ is converted to depth by $z=c\tau /2$; the factor two accounts for outward and return travel. Peaks mark reflecting interfaces along that line. ^[ref]

The horizontal coordinate is depth inferred from travel time, while the vertical coordinate is signal amplitude. An A-mode ultrasound scan therefore provides a one-dimensional depth profile rather than the spatial cross-section assembled by a B-mode ultrasound scan.

For a pure rightgoing acoustic wave, retarded time $\theta =t-x$ and slow distance $X=Mx$ separate fast oscillations from nonlinear steepening. Expanding the equation of state gives $p=\rho +M(\gamma -1){\rho }^2/2+O(M^2)$. The leading perturbations satisfy $u_0={\rho }_0=p_0$. Adding the first-order mass conservation and momentum conservation equations cancels the unknown first corrections and yields the displayed viscous Burgers equation. Inviscid characteristics satisfy $\theta =s-(\gamma +1)F(s)X/2$, $p=F(s)$; loss of invertibility marks shock formation and the end of the smooth inviscid approximation.

A plane-parallel atmosphere with isothermal sound speed $c_s$ in gravity $-{\Omega }^{2}z$ has mass density ${\rho }_0{e}^{-z^2/(2H^2)}$, $H=c_s/\Omega$. Above a rigid base, vertically directed adiabatic fluid perturbations satisfy $w^{\prime \prime }-z{w}^{\prime }/H^2+({\sigma }^2/{\Omega }^2-1)w/(\gamma H^2)=0$ and $w(0)=0$. Set $x=z/H$ and $\lambda =({\sigma }^2/{\Omega }^2-1)/\gamma$. The substitution $y=e^{-x^2/4}w$ converts finite Gaussian-weighted kinetic energy into square integrability for $-y^{\prime \prime }+x^{2}y/4=(\lambda +1/2)y$. Odd extension across the base and the oscillator lowering operator force $\lambda$ to be an odd positive integer. Thus $w_n$ is proportional to the Probabilists' Hermite polynomial ${\mathrm{He}}_{2n+1}(z/H)$. These are vertical acoustic waves; a general stably stratified atmosphere also admits internal gravity waves with horizontal structure.

An acoustically rigid boundary has zero perturbation normal velocity. For a reflected acoustic plane wave, pressure amplitudes add in phase and the reflection coefficient is $+1$. It is the infinite-surface acoustic impedance limit. A pressure-release boundary instead fixes the pressure perturbation to zero and reflects with coefficient $-1$.

A stratified atmosphere can reflect low-frequency acoustic waves while permitting sufficiently high-frequency waves to escape. In a plane-parallel isothermal atmosphere, the angular acoustic cutoff is approximately $c_s/(2H_{\rho })$, where $H_{\rho }$ is the density scale height. Its detailed form depends on the atmospheric profile and on the wave-variable approximation.

In an isothermal atmosphere with constant density scale height $H$, the equation $\delta {\rho }_{tt}-c^2\delta {\rho }_{zz}-(c^2/H)\delta {\rho }_z=0$ loses its first spatial derivative under $\delta \rho =e^{-z/(2H)}\Psi$. Direct differentiation gives ${\Psi }_{tt}-c^2{\Psi }_{zz}+c^2\Psi /(4H^2)=0$. Thus a plane wave has dispersion relation ${\omega }^2=c^2{k}^2+{\omega }_c^2$, with real acoustic cutoff frequency ${\omega }_c=c/(2H)$. The cutoff term is positive; a minus sign would instead predict growing sufficiently long spatial modes. Vertical propagation requires $|\omega |\ge {\omega }_c$ within this ideal constant-coefficient model.

A pressure-release boundary imposes zero acoustic pressure disturbance, $\tilde{p}=0$. The incident and reflected pressure amplitudes therefore obey $1+R=0$, so the pressure reflection coefficient is $R=-1$. It contrasts with a rigid inviscid boundary, where zero normal velocity gives $R=+1$. A much lower normal impedance on the far side approximates the pressure-release condition.

A sonic boom is the sound associated with shock waves from a body moving at a supersonic Mach number. The weak-wave Mach cone describes a propagation geometry rather than the finite pressure signature determining the sound strength.