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

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.

Compressing a small region raises its mass density and, for positive compressibility, its pressure. The outward pressure gradient force accelerates neighbouring fluid elements. Their outward motion compresses fluid farther away, carrying the disturbance outward. The original compressed region relaxes; inertia allows the fluid motion to continue beyond zero excess pressure, producing a rarefaction and a restoring pressure gradient force. Elastic compression and inertia therefore support a propagating longitudinal wave.

In a local planar description with acoustic displacement $\xi (x,t)$,so ${\xi }_{tt}=c_0^2{\xi }_{xx}$. The speed of sound is a propagation speed, whereas the acoustic particle velocity is the small back-and-forth motion of the medium. In linear acoustics, energy travels with the disturbance without first-order net transport of the fluid over a wave period.

Use a plane wave proportional to $e^{i(kx-\omega t)}$. Substitution into the convected acoustic wave equation gives the dispersion relationThe two branches have signed phase velocities and group velocities $U_0\pm c_0$. On a branch with $\omega -U_{0}k=s{c}_{0}k$, $s=\pm 1$, the linearized Euler equation givesIn subsonic flow with $0\le U_0<c_0$, the absence of wave reflection selects the outgoing branch on each side:For $U_0=0$, take $A$ real and positive by choosing the source phase, and put $k_0=\omega /c_0$. At $\omega t=2n\pi$, the requested spatial profiles areFor complex $A$, retain $\mathrm{Re}[A{e}^{i{k}_0|x|}]$ instead. The normalized sound pressure and acoustic density perturbation lie on the same even cosine curve. The acoustic particle velocity follows that curve on the right and its negative on the left, as in the figure in part 2(a). Compressed fluid moves away from the source on both sides. These are two outgoing travelling waves, not a standing wave. The ideal point loudspeaker supplies the discontinuity of velocity at $x=0$; the source-free equations apply on either side of it.

Take the wind to blow toward positive $x$, so $U_0\ge 0$. In subsonic flow, both stationary observers record the source angular frequency $\omega$ and, for the imposed density boundary data, the same acoustic density perturbation amplitude ${\rho }_0|A|$. Their spatial records distinguish the wavelengths and directions:The downstream pattern moves at $c_0+U_0$ and the upstream pattern at signed velocity $U_0-c_0$. Thus the downstream observer sees longer wavelengths while the upstream observer sees shorter ones. The interval containing multiple wavelengths lets each measure the spatial spacing and its direction of travel. The arrival delays at $\pm L$ are $L/(c_0+U_0)$ and $L/(c_0-U_0)$ respectively; after arrival their temporal traces differ by these propagation phases, not by temporal frequency.

In supersonic flow, both characteristic curves have positive slopes $dx/dt=U_0\pm c_0$. With an initially undisturbed upstream region, the observer at $-L$ sees no source-generated disturbance. The observer at $+L$ can see two downstream travelling waves, withThe faster signal arrives first, at $L/(U_0+c_0)$; the slower one arrives at $L/(U_0-c_0)$. Once both are present, their different wavenumbers can produce spatial modulation, while the time dependence at a fixed position remains harmonic with angular frequency $\omega$.

There is a qualification to the source data: the single prescribed value of ${\rho }_1(0^{+},t)$ fixes only $A_{+}+A_{-}=A$. It does not determine the two complex amplitudes separately. A second source condition, such as its downstream acoustic particle velocity with amplitude $c_0(A_{+}-A_{-})$, or a specified mass/force source model, is needed. Consequently two downstream branches are allowed; the stated density data alone do not require both to have nonzero amplitude. Extending a formally upstream-going subsonic formula into $x<0$ when $U_0>c_0$ would impose a signal arriving from upstream infinity and violate the causal source interpretation.

As $U_0\to c_0$, one laboratory phase velocity tends to zero. Its wavelength tends to zero, its wavenumber diverges, and its arrival time diverges; a finite-time experiment never attains the assumed established upstream field at fixed $L$. Finite source size, viscosity, thermal losses and eventually molecular scales invalidate this singular inviscid flow limit. With effective acoustic diffusivity ${\nu }_a$, a loss estimate is ${\nu }_a{k}^2/\omega \sim {\nu }_a\omega /(U_0-c_0{)}^2$, which diverges. Moreover the finite-amplitude change in the characteristic speed is of order $c_0|A|$, so uniform linearization requires $|U_0-c_0|\gg c_0|A|$. At exactly sonic background speed, the zero-speed branch cannot support a nonzero-$\omega$ harmonic solution. Nonlinear acoustics and losses must resolve the near-sonic regime; an unlimited spatial compression of the ideal wave is not physical.

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.

The speed of sound is the propagation speed of a small mechanical pressure disturbance through a medium. Compression creates a restoring pressure gradient, while the medium's mass density supplies inertia. For an inviscid fluid undergoing locally isentropic disturbances,A stiffer pressure response increases the speed. For a perfect gas, this becomes $c_s^2=\gamma p/\rho$, with $\gamma$ the heat capacity ratio.

The disturbance travels while individual fluid parcels undergo much smaller oscillatory motions; acoustic particle velocity and propagation speed are different quantities. Timing a short sound pulse between two sensors a known distance apart gives a direct measurement after accounting for electronic delays. If heat exchange maintains a different thermodynamic response during compression, the relevant derivative changes, as in the isothermal sound speed. The formula also presumes a wavelength and timescale for which the continuum model is appropriate.