Nonlinear Wave Propagation in 1D Random Media J. B. Thoo, Yuba College UCI Computational and Applied Math Seminar Winter 22 Typeset by FoilTEX
Background O Doherty and Anstey (Geophysical Prospecting, 1971) looked at the effects of multiple scattering by a pulse propagating through the earth s crust. With the advent of digitally-generated seismograms, they found that including multiple reflections increased the amplitude of the primary pulse enough to offset transmission losses. Idea: Multiple scattering constructive interference. incident wave directly transmitted wave sum of double-scattering ( constructive interference ) UCI Comput. Appl. Math. Seminar, Wtr 2 1
Many contributions since. Burridge, Papanicolaou, White (1988); Burridge and Chang (1989); Clouet and Fouque (1994); Lewicki, Burridge, and Papanicolaou (1994); Sølna (1998); and others. Some to make the O Doherty-Anstey theory rigorous and some to extend the theory to cases where fluctuations in the medium may not be weak. UCI Comput. Appl. Math. Seminar, Wtr 2 2
Overview We study the propagation of a weakly nonlinear wave pulse, governed by a hyperbolic system of conservation laws, through a stationary random medium in 1D. Examples Gas dynamics (interaction of sound waves with background entropy wave). Nonlinear elastic waves in stratified media. MHD (e.g., scattering of magnetoacoustic waves off Alfvén waves). UCI Comput. Appl. Math. Seminar, Wtr 2 3
Scalings Incident pulse (primary wave): amplitude O(ε 2 ), width O(ε 2 ) Fluctuations (background medium or scattering wave): amplitude O(ε), width O(ε 2 ) Propagation: distance O(1/ε 2 ), time O(1).2.4.15.2.1.5 -.5 -.2 -.1 -.4 -.15 -.4 -.2.2.4 -.2 -.25 -.2 -.15 -.1 -.5.5.1.15.2.25 UCI Comput. Appl. Math. Seminar, Wtr 2 4
Expansion We use the asymptotic expansion introduced by Benilov and Pelinovskĭi (1988) to get a nonlinear O Doherty-Anstey theory. The asymptotic solution includes a realization-dependent random phase shift. W.r.t. this reference frame, the pulse moves as a Brownian motion and the leading order solution is nonrandom. This self-averaging of the leading-order solution means that closure problems, u 2 u 2, do not arise in the nonlinear theory. UCI Comput. Appl. Math. Seminar, Wtr 2 5
In the simplest case, the normalized form of asymptotic equation for the wave profile u is u t + ( 1 2 u2) x = k(y)u xx (x y, t)dy, where k(y) = β( )β( + y) is the covariance function of scattering coefficients β (random fluctuations from the double scattering of waves). The wave profile satisfies an inviscid Burgers equation with a nonlocal, lower-order dissipative and dispersive term that describes the effects of double scattering of waves off the fluctuations in the medium or scattering wave. The fluctuations are assumed to be uniformly bounded in space. UCI Comput. Appl. Math. Seminar, Wtr 2 6
Outline Sketch of the derivation of the asymptotic equations Scattering by a nonuniform medium Scattering of one wave off another wave Examples: 1D gas dynamics, nonlinear elasticity Properties of the asymptotic equations (shock profiles, shock formation) UCI Comput. Appl. Math. Seminar, Wtr 2 7
Derivation of asymptotic equations Scattering by a nonuniform medium Quadratic nonlinearity Consider a strictly hyperbolic system of conservation laws of the form u t + [ f(u) + εb(x/ε 2 )u ] x = u = u(x,t) R m defined for < x < and t f : R m R m B = B(x/ε 2 ) R m m is a realization of a stationary random process with zero spatial mean (assumed to be uniformly bounded) ε is a small, positive parameter UCI Comput. Appl. Math. Seminar, Wtr 2 8
Seek an asymptotic solution u ε (ε ) of the form u ε = ε 2 u 2 (ξ,t) + ε 3 u 3 (ξ, y, t) + ε 4 u 4 (ξ, y, t) + O(ε 5 ) y = x ε 2, ξ = x λt ε3 φ(x/ε 2 ) ε 2 φ is a realization of a random process and is to be chosen u 2 is the primary wave UCI Comput. Appl. Math. Seminar, Wtr 2 9
Find that Primary wave is u 2 = a(ξ,t)r Random correction to the pulse location is where ( x ) εφ ε 2 = 1 x/ε 2 λ ε β(y) dy, β = l T B(y)r describes the strength of an i-wave generated by the scattering of a k-wave off the fluctuations of the medium. Remark: φ typically converges in distribution to a Brownian motion. E.g. [Lipster and Shiryaev], if β is strictly stationary with a suitable mixing condition. UCI Comput. Appl. Math. Seminar, Wtr 2 1
The asymptotic equation for the wave profile of the primary wave is a t + ( 1 2 Ga2) = ± Λ W(σ)a ξξ (cσ + ξ,t) dσ, ξ where W(σ) = β(y)β(y σ) and the average is given by Remarks: f(, y, ) = lim L 1 L L f(, y, )dy. G is a coefficient of quadratic nonlinearity that describes the strength of the j-wave generated by quadratic self-interaction. W is the covariance of scattering coefficients. UCI Comput. Appl. Math. Seminar, Wtr 2 11
Wave-wave scattering Conservative form Consider a strictly hyperbolic system of conservation laws u t + f(u) x = u = u(x,t) R m defined for < x < and t f : R m R m UCI Comput. Appl. Math. Seminar, Wtr 2 12
Seek asymptotic solution u ε (ε ) of the form u ε = εu 1 (ξ,t) + ε 2 u 2 (ξ,η, t) + ε 3 u 3 (ξ, η, t) + ε 4 u 4 (ξ, η, t) + O(ε 5 ) ξ = x λt ε3 φ(η, t) ε 2, η = x µt ε 2 φ is a realization of a random process to be chosen u 1 is the background scattering wave and u 2 is the primary wave UCI Comput. Appl. Math. Seminar, Wtr 2 13
A similar derivation as before leads us to find that the random correction to the pulse location is εφ(η, t) = 1 η λ µ Γ s(y, t) dy, where Γ describes the strength of an i-wave produced by the quadratically nonlinear interaction of a k-wave with a p-wave. UCI Comput. Appl. Math. Seminar, Wtr 2 14
The asymptotic equations for the wave profile of the background scattering wave s = s(η, t) and for the wave profile of the primary wave a = a(ξ, t) are s t + ( 1 3 Hs3) η =, (a + Z s 2 ) t + ( 1 2 Ga2) ξ = ± Λ W(σ, t)a ξξ (ξ cσ, t) dσ, where W(σ, t) = s(η, t)s(η σ, t). UCI Comput. Appl. Math. Seminar, Wtr 2 15
Remarks: s is assumed to be a stationary random process with zero spatial mean. s is not genuinely nonlinear at u =. G is a coefficient of quadratic nonlinearity that describes the strength of the j-wave generated by quadratic self-interaction. Λ, the coefficients of the nonlocal terms, describe the strength of the j-wave generated by double scattering: j p k, k p j. W is a covariance of the background scattering wave. We get the same asymptotic equations with slightly modified coefficients for the system of hyperbolic conservation laws in nonconservative form u t + A(u)u x =. UCI Comput. Appl. Math. Seminar, Wtr 2 16
Example: 1D gas dynamics Scattering of a nonlinear sound wave off entropy fluctuations 1D compressible Euler equations ( ρe + 1 2 ρv2) t + p = κρ γ exp(s/c v ), c 2 = γp ρ ρ t + (ρv) x =, (ρv) t + (ρv 2 + p) x =, ( ρev + 1 ) 2 ρv3 + pv = uniform background state with density ρ, velocity v =, entropy S, and sound speed c = γp /ρ x UCI Comput. Appl. Math. Seminar, Wtr 2 17
This leads to the asymptotic equations (in normalized form) for the wave profiles of the entropy wave (background scattering wave) s t = right-moving sound wave (primary wave) a t + ( 1 2 a2) x = W ( σ 2) a xx (x σ, t) dσ, where W(σ) = s(y)s(y σ). Remark: The entropy wave is not genuinely nonlinear at u = ; in fact, it is linearly degenerate. UCI Comput. Appl. Math. Seminar, Wtr 2 18
So, the asymptotic solution of the compressible Euler equations is where ρ v S ρ S + ε 2 a(ξ, t) + εs(η) ρ c p /c v c 2 + O(ε 3 ), η = x ε 2, ξ = x c t ε 3 φ(x/ε 2 ) ε 2 φ(x/ε 2 ) = c2 2γc v x/ε 2 s(y) dy. UCI Comput. Appl. Math. Seminar, Wtr 2 19
Example: Nonlinear elasticity Elastic wave propagating in 1D through an isotropic hyperelastic medium Deformation x 1 = x + u 1 (x, t), x 2 = x 2 + u 2 (x,t) Equations of motion ρu 1tt = (g(u 1x, u 2 2x)) x, ρu 2tt = (u 2x h(u 1x, u 2 2x)) x density ρ: R R strain-energy density σ: R R R constitutive functions g, h: R R R g(p, q) = σ (p, q), p h(p, q) = 2 σ(p, q) q UCI Comput. Appl. Math. Seminar, Wtr 2 2
Scattering of longitudinal waves in a nonuniform medium Assume the Lamé moduli and density have the expansions (ε ) λ(x) = λ + ελ 1 (x/ε 2 ) + O(ε 2 ), µ(x) = µ + εµ 1 (x/ε 2 ) + O(ε 2 ), ρ(x) = ρ + ερ 1 (x/ε 2 ) + O(ε 2 ). Then we can write the equations of motion in the form u t + [ f(u) + εb(x/ε 2 )u ] x =, where B = (2µ 1 + λ 1 ) µ 1 ρ 1 /ρ 2 ρ 1 /ρ 2. UCI Comput. Appl. Math. Seminar, Wtr 2 21
This leads to the asymptotic equation (in normalized form) for the wave profile of the right-moving longitudinal wave a t + ( 1 2 a2 ) x = W ( σ 2) a xx (x σ, t) dσ, where W(σ) = β + (y)β (y σ), β ± (y) = 1 2ρ c l [ 2µ1 (y) + λ 1 (y) ± c 2 l ρ 1 (y) ] and c l is the longitudinal wave speed. Note: The nonlocal term on the RHS originates from the scattering of a right-moving longitudinal wave into a left-moving longitudinal wave. The scattering of a longitudinal wave does not produce any transverse waves, and vice versa, in an isotropic medium. UCI Comput. Appl. Math. Seminar, Wtr 2 22
So, the asymptotic solution of the equations of motion is where m 1 m 2 w 1 w 2 ε2 a(ξ, t) φ(x/ε 2 ) = 1 2ρ c l c t 1 1/(ρ c l ) m k = ρu kt, w k = u kx, ξ = x c tt ε 3 φ(x/ε 2 ) ε 2, x/ε 2 and c t is the transverse wave speed. + O(ε3 ), [ 2µ1 (y) + λ 1 (y) c 2 l ρ 1 (y) ] dy UCI Comput. Appl. Math. Seminar, Wtr 2 23
Scattering of transverse waves in a nonuniform medium The asymptotic equation (in normalized form) for the wave profile of the right-moving transverse wave is a t ( 1 3 a3) x = V ( σ 2) a xx (x σ, t) dσ, where V (σ) = β + (y)β (y σ), β ± (y) = 1 2ρ c t [ µ1 (y) ± c 2 tρ 1 (y) ]. UCI Comput. Appl. Math. Seminar, Wtr 2 24
The asymptotic solution of the equations of motion is where m 1 m 2 w 1 w 2 εa(ξ, t) φ(x/ε 2 ) = 1 2ρ c l c t 1 1/(ρ c t ) m k = ρu kt, w k = u kx, ξ = x c lt ε 3 φ(x/ε 2 ) ε 2, x/ε 2 + O(ε2 ), [ µ1 (y) c 2 tρ 1 (y) ] dy. UCI Comput. Appl. Math. Seminar, Wtr 2 25
Scattering of longitudinal wave off transverse wave Density, ρ = ρ, Lamé moduli λ = λ and µ = µ The asymptotic equations (in normalized form) for the wave profiles of the right-moving transverse wave (background scattering wave) s t + ( 1 3 Hs3) η =. right-moving longitudinal wave (primary wave) a t + ( 1 2 a2) ξ = W ( σ c, t ) a ξξ (ξ σ, t)dσ W(σ, t) = s(η, t)s(η σ, t). Remark: The transverse waves are not genuinely nonlinear at u =. UCI Comput. Appl. Math. Seminar, Wtr 2 26
The asymptotic solution of the equations of motion is m 1 m 2 w 1 w 2 εs(η, t) 1 1/(ρ c t ) + ε 2 a(ξ,t) 1 1/(ρ c l ) + O(ε3 ), where m k = ρu kt, w k = u kx, η = x c tt ε 2, ξ = x c lt ε 2. UCI Comput. Appl. Math. Seminar, Wtr 2 27
Properties of the asymptotic equations Shock profiles In the simplest case, the normalized form of asymptotic equation for the wave profile u is u t + ( 1 2 u2) x = k(y)u xx (x y, t) dy, where k is the covariance function of the random fluctuations. The right-hand side describes the effects of double scattering of the incident pulse wave. is a lower-order dissipative and dispersive term, and its dissipative effects may prevent the formation of shocks. UCI Comput. Appl. Math. Seminar, Wtr 2 28
Example 1 k(x) = exp( x ) random telegraph process Ornstein-Uhlenbeck process Applying the operator x + 1 to the normalized equation yields the relaxing gas equation, [ ( x + 1) u t + u x + ( 1 2 u2 ) x ] u x =. Traveling wave solution u = u(x ct, t) satisfies (c 1 u)u + 1 2 u2 cu =. lim x = u L, lim x u = c = u L /2 UCI Comput. Appl. Math. Seminar, Wtr 2 29
If u L > 2 (c > 1), then the solution is smooth with an embedded subshock with shock speed c = 1 + u. u = 2c L 2(c 1) c = 1 + <u> u = R If < u L < 2 ( < c < 1), then the solution has a smooth wave profile. UCI Comput. Appl. Math. Seminar, Wtr 2 3
Example 2 k(x) = { 1 x if x 1, if x > 1. This leads to the partial differential-difference equation u t + ( u + 1 2 u2) x = u(x + 1, t) u(x,t). The linearized equation has solution u t + u x = u(x + 1, t) u(x, t) u(x, t) = n= t n n! e t f(x t + n), u(x,) = f(x). So, an initial wave profile splits into multiple pulses as the wave propagates. UCI Comput. Appl. Math. Seminar, Wtr 2 31
1.5 q(1) at time t = 1.5 q(1) at time t = 1 1 1.5.5.5 6 5 4 3 2 1 1 2.5 6 5 4 3 2 1 1 2 1.5 q(1) at time t = 2 1.5 q(1) at time t = 3 1 1.5.5.5 6 5 4 3 2 1 1 2.5 6 5 4 3 2 1 1 2 u t + ( 1 2 u2) x = u(x + 1, t) u(x, t), u(x,) = exp( 25x 2 ) UCI Comput. Appl. Math. Seminar, Wtr 2 32
Shock formation For appropriate covariance functions k, if a smooth pulse satisfying u t + ( 1 2 u2 ) x = k(y)u xx (x y, t) dy is nowhere too steep when t =, then it will not break; and if the pulse is anywhere too steep at t =, then it will break in finite time. Of importance is the condition that k ( ) >, for it provides a damping term, evident after integration by parts twice: u t + (k( ) + u)u x = k ( )u + k (y)u(x y, t) dy. The integral term alone on the RHS may not be enough to prevent wave breaking. UCI Comput. Appl. Math. Seminar, Wtr 2 33
Theorem. Assume that 1. k H 3 (R) is a covariance function and k ( ) >. 2. u = u(x, t) is a smooth solution of u t + ( 1 2 u2) x = k(y)u xx (x y, t)dy that decays rapidly as x ±. Let α = k ( ) and C = k ( ) u + ( 1 α k ( ) k 2 + k 2 ) u 2. 1. If C < α 2 /4 and u x > (α + α 2 4C)/2 for all x R, then the solution u does not break. 2. If u x < (α + α 2 + 4C)/2 for some x R, then the solution breaks in finite time. In fact, u x as t [ (α + α 2 + 4C)/2 u x ] 1. UCI Comput. Appl. Math. Seminar, Wtr 2 34
Sketch of proof energy estimate u 2 u 2 sup-norm estimate u u e αt + 1 α k 2 u 2 α, C R + and C dv dt + v2 + αv C v is globally smooth if α 2 4C > and v() > (α + α 2 4C)/2 v blows up in finite time if v() < (α + α 2 + 4C)/2 UCI Comput. Appl. Math. Seminar, Wtr 2 35