Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 335 / 2

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 $G_L(\mathbf{\zeta },\mathbf{z})=G(L,\mathbf{\zeta };0,\mathbf{z})$, with the chosen source and array normalizations incorporated into the Green function. Let $a(\mathbf{\zeta })\ge 0$ be the array's aperture weight, equal to the indicator of its receiving region for an ideal uniform array. The recorded field isBy wave reciprocity, back-propagation has the same Green function with the source and receiver exchanged. Therefore the physically re-emitted, back-propagated wave amplitude isIf $Hf=R$ and $P_A$ multiplies by $a$, then ${\Psi }_B=H^T{P}_A\overline{Hf}$, where $T$ denotes the transpose without conjugation. Taking a final complex conjugate instead defines the adjoint reconstruction $\overline{{\Psi }_B}=H^{\ast }{P}_{A}Hf$. 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. Random scattering can improve finite-aperture wave focusing through angular diversity. See the regime-dependent analysis in Statistical stability in time reversal.

Fix a receiver position $\mathbf{\zeta }$ and define $g_{\mathbf{\zeta }}(\mathbf{z})=G_L(\mathbf{\zeta },\mathbf{z})$. Choose the two scalar fields in the Wigner distribution to be $u=v=g_{\mathbf{\zeta }}$. Fourier inversion in the separation variable givesIndeed, substituting the defining Wigner distribution makes the $\mathbf{p}$ integral a $(2\pi {)}^2$ Dirac delta function, canceling its normalization and setting the separation equal to $\mathbf{z}-{\mathbf{z}}^{\prime }$.

Average over the receiver plane, including the aperture:Then the receiver-averaged Wigner distribution represents the back-propagated field asThe fast decay of $f$ justifies its use as a test function. For oscillatory Green functions that are not integrable, the Wigner distribution and its inversion can be understood as tempered distributions, or derived with smooth cutoffs before taking their limits.

The printed $W_A$ expression can also be used literally, but its displayed ${\mathbf{z}}^{\prime }$ integral is over source-plane coordinates, not receiver positions. The missing receiver average and source weight can be encoded by choosing vector fields whose channels are the receivers. LetContract the receiver channels in the product of the vector fields, so thatWith exactly the printed definitionthe same Fourier inversion now yields the requested compact formFor a finite array, the receiver integral in the contraction is a weighted sum. For a continuous array, these are fields valued in the receiver Hilbert space. An uncontracted vector outer product instead gives a matrix-valued Wigner distribution, whose receiver trace must be taken. Thus the printed formula is usable with these choices and this contraction convention, but its phrase “over the plane of the receiver” does not describe the displayed integral.

Let $H$ be the measured wave scattering response> linear operator, with direct background propagation removed if present. Its time reversal operator is the positive operator $T=H^{\ast }H$. For a normalized emitted signal $h$,The maximizing signal is a right singular vector, equivalently an eigenvector of $T$ with its largest eigenvalue. This is a precise intensity statement independent of a scatterer model.

Under the Born approximation for scalar wave scattering, neglect interactions between point scatterers and writewhere $b_j$ is the receiver response to scatterer $j$, $c_j^{\ast }h$ is its illumination by the source array, and ${\alpha }_j$ is its scattering wave amplitude. Well-resolved point scatterers have approximately orthogonal vectors $b_j$ and $c_j$. In the ideal orthogonal limit,Thus the most reflective, geometrically weighted scatterer gives the largest eigenvalue, and its normalized steering vector $c_j/\Vert c_j\Vert$ gives the corresponding wave focusing signal. If illumination and reception factors are equal for all scatterers, this is precisely the scatterer with largest $|{\alpha }_j{|}^2$. Geometric size by itself is not the quantity being ranked.

Repeated adjoint time reversal acoustics is power iteration on $T$. If its top eigenvalue is simple and the initial signal has a nonzero component along its eigenvector, thenA tied largest eigenvalue leaves a combination in the leading eigenspace, and an initial signal orthogonal to that eigenspace cannot excite it. This is the basis of the DORT method. Mere physical separation is insufficient if the array cannot resolve the scatterers; coherent steering-vector overlap or multiple scattering can mix the modes. Also, if $H$ includes unrestricted homogeneous transmission, its largest eigenvalue need not identify any individual scatterer. The correspondence concerns the resolved wave scattering response>. The ideal correspondence and selective wave focusing are analyzed by Prada and Fink.