= Solution
{c}
Take the wind to blow toward positive $x$, so $U_0\geq0$. 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:
$$
\boxed{\lambda_{\mathrm{down}}=\frac{2\pi(c_0+U_0)}{\omega},\qquad
\lambda_{\mathrm{up}}=\frac{2\pi(c_0-U_0)}{\omega}.}
$$
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, \b[the observer at $-L$ sees no source-generated disturbance]. The observer at $+L$ can see two downstream <travelling waves>, with
$$
k_+=\frac{\omega}{U_0+c_0},\qquad k_-=\frac{\omega}{U_0-c_0},\qquad
\rho_1=\rho_0\operatorname{Re}\left[e^{-i\omega t}\left(A_+e^{ik_+x}+A_-e^{ik_-x}\right)\right].
$$
The 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.
Back to article page
