# Pulse Propagation and Time Reversal in Shallow-Water Acoustic Random Waveguides

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## High-Frequency Approximation to Coupled Power Equations

In this section we give, under the assumption that nearest neighbor coupling is the main power transfer mechanism, an approximation of the solution of the coupled power equations (2.47) in the high-frequency regime or in the limit of large number of propagating modes N(ω) ≫ 1. Let us note that the limit of a large number of propagating modes N(ω) ≫ 1 corresponds to the high-frequency regime ω → +∞. Next, we analyze the energy carried by the propagating modes in this regime. The coupled power equations can be approximated in the high-frequency regime by a diffusion equation. This approximation has been already obtained in  for instance, in which we can find further references about this topic. We can also refer to  for more discussions on this approximation. For an application of such a diffusion model to acoustic propagation in random sound channels we refer to , and for applications to time reversal of waves we refer to  and Chapters 3 and 4 of this manuscript .

### High-Frequency Approximation to Coupled Power Equation with Negligible Radiation Losses

In the case of negligible radiation losses, we also get a continuous diffusive model for the coupled power equations in the high-frequency regime or in the limit of a large number of propagating modes N(ω) ≫ 1. This diffusive continuous model is equipped with boundary conditions which take into account the negligible effect of the radiation losses at the bottom and the free surface of the waveguide (see Figure 2.4 page 57). Now, let us assume that the radiation losses are negligible, that is, Λc(ω) = τ˜Λc(ω) with τ ≪ 1. We have already remarked that, if the radiation losses are negligible, then the coupling process is predominant and we have ∀L > 0, sup z∈[0,L] kT τ,l j (ω, z) − T 0,l j (ω, z)k2,RN(ω) = O(τ ).

#### Mode Coupling in Random Waveguides

In this section, we study the Fourier transform bp(ω, x, z) of the pressure p(t, x, z) when a random section [0, L/ǫ] is inserted between two homogeneous waveguides. In the half-space z ≥ 0, by taking the Fourier transform in (3.2), we get the perturbed time harmonic wave equation ∂2 z bp(ω, x, z) + ∂2 x bp(ω, x, z) + k2(ω)(n2(x) + √ǫ ˜ V (x, z))bp(ω, x, z) = 0.

1 Introduction
1.1 Propagation des ondes en milieux aléatoires
1.2 Presentation of the results
1.2.1 Wave Propagation
1.2.2 Time reversal
2 Wave Propagation in Shallow-Water Acoustic Random Waveguides
2.1 Waveguide Model
2.2 Wave Propagation in a Homogeneous Waveguide
2.2.1 Spectral Decomposition in Unperturbed Waveguides
2.2.2 Modal Decomposition
2.3 Mode Coupling in Random Waveguides
2.3.1 Coupled Mode Equations
2.3.2 Energy Flux for the Propagating and Radiating Modes
2.3.3 Influence of the Evanescent Modes on the Propagating and Radiating Modes
2.3.4 Forward Scattering Approximation
2.4 Coupled Mode Processes
2.4.1 Limit Theorem
2.4.2 Mean Mode Amplitudes
2.5 Coupled Power Equations
2.5.1 Exponential Decay of the Propagating Modes Energy
2.5.2 High-Frequency Approximation to Coupled Power Equations
2.5.3 High-Frequency Approximation to Coupled Power Equation with Negligible
2.6 Appendix
2.6.1 Gaussian Random Field
2.6.2 Proof of Theorem 2.1
2.6.3 Proof of Theorem 2.2
2.6.4 Proof of Theorem 2.4
2.6.5 Proof of Theorem 2.6
3 Pulse Propagation and Time Reversal in Shallow-Water Acoustic Random Waveguides
3.1 Waveguide Model
3.2 Mode Coupling in Random Waveguides
3.2.1 Coupled Mode Equations
3.2.2 Propagator and Forward Scattering Approximation
3.2.3 Limit Theorem
3.3 Pulse Propagation in Random Waveguides
3.3.1 Broadband Pulse in Homogeneous Waveguides
3.3.2 Broadband Pulse in Random Waveguides
3.3.3 Incoherent Fluctuations in the Broadband Case
3.4 Time Reversal in a Waveguide
3.4.1 First Step of the Experiment
3.4.2 Second Step of the Experiment
3.4.3 Refocused Field in a Homogeneous Waveguide
3.4.4 Limit Theorem
3.4.5 Refocused Field in a Changing Random Waveguide
3.4.6 Stability of the Refocused Wave
3.4.7 Mean Refocused Field in the Case μ → 1
3.4.8 Numerical Illustration
4 Time Reversal SuperResolution in Random Waveguides
4.1 Waveguide Model
4.2 Waveguide Propagation
4.2.1 Propagation in Homogeneous Waveguides
4.2.2 Mode Coupling in Random Waveguides
4.2.3 Band-Limiting Idealization and Forward Scattering Approximation
4.3 The Coupled Mode Process
4.4 Time Reversal in a Waveguide
4.4.1 First Step of the Time Reversal Experiment
4.4.2 Second Step of the Time-Reversal Experiment
4.4.3 Homogeneous Waveguide
4.4.4 Mean Refocused Field in the Random Case
4.4.5 Statistical Stability
4.4.6 Quarter Wavelength Plate
4.5 Appendix
4.5.1 Proof of Theorem 4.1
4.5.2 Proof of Theorem 4.2
Bibliography

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