Past exam of the fluids course of the University of Oxford / 2021 / 2

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.

For one-dimensional inviscid flow, conservation of mass and the Euler equation areSubstitute $u=U_0+v_1$ and the stated background-plus-perturbation expressions. Dropping products of perturbations gives the linearizationThe Taylor expansion of the equation of state givesSince the background coefficients are constant, $D_0$ commutes with ${\partial }_x$. Applying $D_0$ to the linearized continuity equation and using the linearized Euler equation therefore yieldsThis is the convected acoustic wave equation: in the coordinate $x-U_{0}t$, it becomes the ordinary one-dimensional wave equation. Positive $c_0^2$ is required for propagating sound waves.

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.