Shock structure due to the stochastic forcing of waves Shock structure due to the stochastic forcing of waves Andr´ e Nachbin IMPA/BRAZIL HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves Motivation: WAVEFORM INVERSION/Refocusing A FANTASTIC APPLICATION! 2D linear HYPERBOLIC waves ⇒ 1D nonlinear waves HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves DUSAN PETRICIC TIME-REVERSED ACOUSTICS Arrays of transducers can re-create a sound and send it back to its source as if time had been reversed. The process can be used to destroy kidney stones, detect defects in materials and communicate with submarines by Mathias Fink I n a room inside the Waves and Acoustics Laboratory in the loudspeakers, the sound of the “olleh” converges onto Paris is an array of microphones and loudspeakers. If your mouth, almost as if time itself had been reversed. In- you stand in front of this array and speak into it, any- deed, the process is known as time-reversed acoustics, and thing you say comes back at you, but played in reverse. Your the array in front of you is acting as a “time-reversal mirror .” “hello” echoes — almost instantaneously — as “olleh.” At first Such mirrors are more than just a novelty item. They have this may seem as ordinary as playing a tape backward, but a range of applications, including destruction of tumors and there is a twist: the sound is projected back exactly toward kidney stones, detection of defects in metals, and long- HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin its source. Instead of spreading throughout the room from distance communication and mine detection in the ocean.
Shock structure due to the stochastic forcing of waves Acoustic chamber ACOUSTIC SOURCE ELECTRONIC PLAYBACK RECORDINGS OF SIGNALS IN REVERSE SARAH L. DONELSON PIEZOELECTRIC TRANSDUCERS HETEROGENEOUS MEDIUM RECORDING STEP TIME-REVERSAL AND REEMISSION STEP ACOUSTIC TIME-REVERSAL MIRROR operates in two In the second step ( right ), each transducer plays back its sound steps. In the first step ( left ) a source emits sound waves ( orange ) signal in reverse in synchrony with the other transducers. The that propagate out, perhaps being distorted by inhomogeneities original wave is re-created, but traveling backward, retracing its in the medium. Each transducer in the mirror array detects the passage back through the medium, untangling its distortions sound arriving at its location and feeds the signal to a computer. and refocusing on the original source point. HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin They can also be used for elegant experiments in pure physics. back on exactly the reversed trajectory, which again would
Shock structure due to the stochastic forcing of waves Solitary wave: Fouque, Garnier, Mu˜ noz & N., PRL ’04 REFLECTED SIGNAL TRANSMITED SIGNAL INITIAL SOLITARY WAVE −200 −150 −100 −50 0 50 100 150 200 ξ REFOCUSED PROFILE −200 −150 −100 −50 0 50 100 150 200 ξ HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves In order to understand NONLINEAR PDEs with TIME-REVERSED data First we address the DIRECT NONLINEAR SCATTERING PROBLEM ⇒ NonLin Hyperbolic PDEs with HIGHLY VARIABLE coefficients HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves PHYSICAL MODEL: Long propagation distances + detailed TOPOGRAPHY Scientific American ’99 HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves α = nonlinearity = amplitude/depth β = dispersion = depth/wavelength γ = disorder /wavelength h ( x ) ≡ DISORDERED TOPOGRAPHY PROFILE HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves Reflection-Transmission of waves & Time-reversal of waves ...in the diffusion approximation regime: ∼ Acoustics (a) Linear Hyperbolic: ( α = β = 0) (b) Linear Dispersive: ( α = 0; β = ε ) (c) Nonlinear Hyperbolic: ( α = ε ; β = 0) (d) Convection-diffusion: ( α = ε ; µ = ε ) (e) Solitary waves: ( α = β = ε ) HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves OVERVIEW of RESULTS and THEORY HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves SETUP for THEORY and SIMULATIONS: Typical wave profiles: Gaussian, dGaussian/dx and Solitary wave. 250 200 150 100 50 −10 0 10 20 30 40 50 250 200 150 100 50 −2 −1.5 −1 −0.5 0 0.5 1 1.5 1 x 10 −3 0.8 TIME−REVERSED WAVE → 0.6 0.4 0.2 0 −0.2 ← REFLECTED WAVE → TRANSMITTED WAVE −0.4 −0.6 −0.8 RANDOM MEDIUM HALF−SPACE −1 −50 −40 −30 −20 −10 0 10 20 30 40 50 HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves INVISCID NonLinear Shallow Water system w/ a dGaussian/dx pulse 2x2 CONSERVATION LAW with DISORDERED variable COEFFICIENTS Fouque, Garnier & N., Physica D ’04 . RANDOM Forcing ⇒ shock structure: ASYMPTOTICS ⇒ wave elevation ≡ η ( x , t ) governed by VISCOUS Burgers’ "Apparently viscous" profile at t = 6.25(1.25)15.0 0.01 α = 0.004; ε = 0.01 0.005 0 −0.005 −0.01 14.8 15 15.2 15.4 15.6 15.8 16 16.2 16.4 16.6 "Inviscid Burgers" profile at t = 6.25(1.25)15.0 0.01 Initial profile centered at x = 10 0.005 0 −0.005 −0.01 15 15.2 15.4 15.6 15.8 16 16.2 16.4 16.6 16.8 17 x ( All waves centered about solution at t= 6.25) HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves MATH TOOL: a LIMIT THEOREM for Stochastic ODEs Khasminskii’s Theorem (*): consider the IVP ω ∈ (Ω , A , P ) dx ε dt = ε F ( t , x ε ; ω ) , x ε (0) = x 0 and dy d τ = F ( y ) , y (0) = x 0 , where F ( t , · ; ω ) is a stationary process, ergodic... with Z T 1 F ( x ) ≡ lim E { F ( t , x ; ω ) } dt . T →∞ T 0 Then E {| x ε ( t ) − y ( t ) |} ∼ √ ε sup on the time scale 1 /ε. 0 ≤ t (*) R.Z. Khasminskii , On stochastic processes defined by differential equations with a small parameter, Theory Prob. Applications, Volume XI (1966), pp.211-228. R.Z. Khasminskii , A limit-theorem for the solutions of differential equations with random right-hand sides, Theory Prob. Applications, Volume XI (1966), pp.390-406. HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves Setting up Shallow Water Eqn. for Khasminskii’s theorem... ...include viscosity µ = ε 2 µ 0 ... ∂η ∂ t + ∂ (1 + ε h + αη ) u = 0 , ∂ x ∂ x = µ∂ 2 u ∂ u ∂ t + ∂η ∂ x + α u ∂ u ∂ x 2 . With the underlying Riemann Invariants, to leading order... � A � A � 1 � � A − ε h ′ � � � ∂ Q ( x ) ∂ 1 = ∂ x B ∂ t B 2 1 1 B � 3 A + B � ∂ � A � + ε 2 α 0 0 4 0 A + 3 B ∂ t B � 1 � ∂ 2 � A + ε 2 µ 0 � 1 + O ( ε 3 ) , 1 1 ∂ t 2 B 2 ...and using a Lagrangian frame ⇒ random ODE-like setting. HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves Khasminskii’s theorem ⇒ The front pulse/right Riemann Inv. B ε ( x , τ ) := B ( x /ε 2 , τ + x /ε 2 ) converges to ˜ B ! p b 0 (0) W x − φ 0 (0) B ( x , τ ) = ˜ ˜ x , τ − √ . B 0 x 2 2 where ˜ B 0 satisfies the deterministic Burgers equation ∂ ˜ ∂ ˜ B 0 B 0 + 3 α 0 B 0 ∂ x = L ˜ ˜ B 0 ∂τ , 4 � x ˜ c − 1 ( s ) ds . B 0 (0 , τ ) = f ( τ ) , τ ≡ t − z , z ≡ 0 L can be written explicitly in the Fourier domain as Z ∞ « Z ∞ „ µ 0 ω 2 + b 0 (2 ω ) ω 2 L B ( τ ) e i ωτ d τ = − B ( τ ) e i ωτ d τ. 2 4 −∞ −∞ PhysFlu, May 2006 ⇒ EDDY VISCOSITY Garnier & N., PRL 2004, Z ∞ b 0 ( ω ) = E [ h (0) h ( x )] exp( i ω x ) dx 0 HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves Direct SWE numerics versus effective Burgers equation GAUSSIAN WAVE PROFILE 6 x 10 −3 5 4 3 2 1 0 −1 −4 −2 0 x 6 x 10 −3 5 4 3 2 1 0 −1 −1 −0.5 0 0.5 1 x HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
Shock structure due to the stochastic forcing of waves DETERMINISTIC PROFILES with RANDOM ARRIVAL TIMES Impulse responses based on data and analysis 0.1 transmitted downgoing pulse 0.08 0.06 0.04 0.02 0 −0.02 0 5 10 15 20 25 30 35 distance from leading front [m] Impulse responses based on data and analysis 0.1 transmitted downgoing pulse 0.08 0.06 0.04 0.02 0 −0.02 0 2 4 6 8 10 12 14 distance from front [m] HYPERBOLIC PDEs/Lyon, 2006 Andr´ e Nachbin IMPA http://www.impa.br/ ∼ nachbin
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