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

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.