= Solution
In <time reversal acoustics>, the array records the incoming signal, reverses each recorded time trace, and re-emits it through the same medium. This is <phase conjugation> in the frequency domain. For the <time-harmonic wave> convention $e^{-i\omega t}$, reversing a real time trace replaces its positive-frequency <wave amplitude> by its <complex conjugate>. The medium must remain unchanged between recording and re-emission.
Let $f(\mathbf z)=\Psi_0(0,\mathbf z)$ and $G_L(\boldsymbol\zeta,\mathbf z)=G(L,\boldsymbol\zeta;0,\mathbf z)$, with the chosen source and array normalizations incorporated into the <Green function>. Let $a(\boldsymbol\zeta)\geq0$ be the array's aperture weight, equal to the indicator of its receiving region for an ideal uniform array. The recorded field is
$$
R(\boldsymbol\zeta)=\int_{\mathbb R^2}G_L(\boldsymbol\zeta,\mathbf z')f(\mathbf z')\,d\mathbf z'.
$$
By <wave reciprocity>, back-propagation has the same <Green function> with the source and receiver exchanged. Therefore the physically re-emitted, back-propagated <wave amplitude> is
$$
K_A(\mathbf z,\mathbf z')=\int_{\mathbb R^2}a(\boldsymbol\zeta)G_L(\boldsymbol\zeta,\mathbf z)\overline{G_L(\boldsymbol\zeta,\mathbf z')}\,d\boldsymbol\zeta.
$$
$$
\boxed{\Psi_B(0,\mathbf z)=\int_{\mathbb R^2}K_A(\mathbf z,\mathbf z')\overline{f(\mathbf z')}\,d\mathbf z'.}
$$
If $Hf=R$ and $P_A$ multiplies by $a$, then $\Psi_B=H^TP_A\overline{Hf}$, where $T$ denotes the transpose without conjugation. Taking a final <complex conjugate> instead defines the adjoint reconstruction $\overline{\Psi_B}=H^*P_AHf$. This distinction prevents an erroneous conjugation in the <time reversal operator>.
For a localized <Gaussian beam> or <acoustic point source> in a homogeneous medium, a finite <aperture> admits a limited range of angles. The focal width is of order $\lambda L/A$ when $A$ denotes the aperture diameter. In a <random medium>, <multiple scattering> creates paths with a larger angular spread. Each reversed path retraces its route, and the paths interfere constructively at the source. This can produce a larger <effective aperture in time reversal> and a narrower focus, even though the unreversed field has a complicated <speckle pattern>.
This comparison concerns a homogeneous reference medium; a deterministic heterogeneous medium can also provide useful <multipath propagation>. Suitable scale limits or frequency and spatial averaging can make refocusing <self-averaging>. Such <self-averaging> is not automatic for every monochromatic source and every random realization. <Wave absorption>, changing medium parameters, unresolved paths or poor array coverage can spoil refocusing. With complete capture of the propagating modes and a lossless <unitary operator> $H$, ideal adjoint reconstruction is already exact in either medium. \b[Random scattering can improve finite-aperture <wave focusing> through angular diversity.] See the regime-dependent analysis in https://arxiv.org/abs/cond-mat/0206095[Statistical stability in time reversal].
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