LINEAR DISPERSIVE WAVES - PDF Free Download

LINEAR DISPERSIVE WAVES Nathaniel Egwu December 16, 2009

Outline 1 INTRODUCTION Dispersion Relations Definition of Dispersive Waves 2 SOLUTION AND ASYMPTOTIC ANALYSIS The Beam Equation General Solution by Fourier Integrals Asymptotic Behaviour 3 GROUP VELOCITY AND ENERGY PROPAGATION Kinematics Derivation of Group Velocity Extensions to Higher Dimension Energy Propagation 4 Conclusion

Dispersion Relation Ansatz: κ, ω, A are constants. For κ, ω, is the dispersive relation. are called modes. ϕ(x, t) = Ae (iκx iωt), G(ω, κ) = 0 ω = W(κ), Re(ϕ) = A cos(κx ωt + η), η = arg(a), θ = κx ωt determines Re(ϕ).

Dispersion Relation Hence, The phase velocity: normal to κ. We need for W (κ) real. In 1D θ x = κ, θ t = ω, λ = 2π κ, τ = 2π ω. c = ω κ, ω κ determinant 2 W κ i κ 0, j W (κ) 0.

Dispersion Relation Examples: Klein-Gordon (quantum theory): ϕ tt α 2 2 ϕ + β 2 ϕ = 0, ω = ± α 2 κ 2 + β 2 ; Korteweg-de Vries equation (Long water waves): ϕ t + αϕ x + βϕ xxx = 0, ω = ακ βκ 3 ; Boussinesq equation (Longitudinal waves for elasticity): ϕ tt α 2 2 ϕ = β 2 2 ϕ tt, ω = ± ακ 1 + β 2 κ 2 ;

Dispersive Relations The Beam equation with modes ω = ±γκ 2. Generally, ϕ tt + γ 2 ϕ xxxx = 0, ω 2 γ 2 κ 4 = 0, ( p t, x 1, and P a polynomial. Hence, x 2, ) = 0, x 3 x j iκ j, t iω. and P( iω, iκ 1, iκ 2, iκ 3 ) = 0, t iω, x j iκ j.

Definition of Dispersive Waves A linear dispersive wave satisfies and ϕ(x, t) = Ae iκx iωt, ω = W (κ), determinant 2 W κ i κ 0. j Hence when oscillations in space are coupled with oscillations in time through a dispersion relation, we expect typical effects of Dispersive Waves.

The Beam Equation Consider ϕ tt + γ 2 ϕ xxxx = 0, recall ω 2 γ 2 κ 4 = 0, ω = ±γκ 2, From ϕ(x, t) = F(κ) depends on ϕ, ϕ t. F(κ)e (iκx iw (κ)t) dκ,

General Solution by Fourier Integrals Solution: with ϕ(x, t) = we get F 1 (κ)e (iκx iw (κ)t) + F 2 (κ)e (iκx+iw (κ)t) dκ ϕ = ϕ 0 (x), ϕ t = ϕ 1 (x). and ϕ 0 (x) = ϕ 1 (x) = i (F 1 (κ) + F 2 (κ))e iκx dκ, W (κ)(f 1 (κ) F 2 (κ))e iκx dκ.

Solution by Fourier Integrals Inverse formula: F 1 (κ) + F 2 (κ) = Φ 0 (κ) = 1 2π iw (κ) [F 1 (κ) F 2 (κ)] = Φ 1 (κ) = 1 2π which gives F 1 (κ) = 1 2 ϕ 0 (x)e iκx dx. ϕ 1 (x)e iκx dx. [ Φ 0 + iφ ] 1(κ), F 2 (κ) = 1 [ Φ 0 iφ ] 1(κ) W (κ) 2 W (κ) }. Since ϕ 0 (x), ϕ 1 (x) are real Φ 0 ( κ) = Φ 0 (κ), Φ 1( κ) = Φ 1 (κ), F 1 ( κ) = F1 (κ), F 2( κ) = F2 (κ), if W ( κ) = W (κ), F 1 ( κ) = F2 (κ), F 2( κ) = F1 (κ), if W ( κ) = W (κ).

Asymptotic Behaviour From ϕ(x, t) = x/t fixed, t, ϕ(x, t) = around κ = k and F(κ)e iκx iw (κ)t dκ, F(κ)e iχt dκ, χ = W (κ) κ x t. χ (κ) = W (κ) x t = 0. By Taylor expansion of χ(κ) and F(κ), F(κ) F(k), χ(κ) = χ(k) + (κ k) 2 χ (κ), χ (κ) 0.

Asymptotic Behaviour We obtain from ϕ F (k)e iχ(κ)t ( exp i ) 2 (κ k)2 χ (κ)t dκ, e αz2 dz = π α, (path through ±π/4), 2π ϕ F(k) t χ (k) exp iχ(k)t i π 4, for Beam equation.

Asymptotic Behaviour From above, k(x, t) : W (k) = x t, k > 0, x t > 0, W (k) = W (k),, If we need χ (κ) 0, and W (k) = const, W (κ) = 0, ϕ F(k)e iχ(k)t exp( i 6 χ (κ)t(κ k) 3 )dκ. So W (κ) = 0, κ = const, equation is undefined on k k(x/t) x/t = W (k).

Kinematic Derivation of Group Velocity Generally, for (x, t), θ(x, t) = xk(x, t) tω(x, t), k = θ x, ω = θ t, k t + ω = 0, t k(x, t) (density) and ω (flux). For ω = W (k), k t + C(k) k x = 0, C(k) = W (k), C(k), for some k, if k = f (x), t = 0, k = f (ξ), x = ξ + C(ξ)t, C(ξ) = C(f (ξ)). gives k from (1), and x = C(k)t, C(k) is the group velocity.

Kinematics Derivation of Group Velocity Hence, for a k 0, x = W (k 0 )t, is the group velocity. For θ(x, t) = θ 0, W (k) = dω dk dx θ x dt + θ t = 0 dx dt defines the phase velocity. = θ t θ x = ω k

Example

Kinematics Derivation of Group Velocity Examples: 1) Beam Equation gives: and W (k) = γk 2 W (k) = 2γk = x t, k = Group and phase lines: x 2γt, ω = x 2 4γt 2, θ = x 2 4γt. c g = 2γk, c p = γk c g > c p. x 2γt = const, x 2 4γt = const.

Kinematics Derivation of Group Velocity 2) Deep water waves: W (k) = gk W (k) = g 4k, k = gt2 4x 2, implies c g < c p. where ω = gt2 2x, θ = gt2 4x c g = 1 2 g g k, c p = k. Thus, gt 2 4x = const, x 2γt = const.

Output for Group and Phase Velocities

Extension to Higher Dimensions General solution: ϕ = F(κ)e iκ.x iw(κ)t dκ, R n ( ) n ( ) 2π 2 F(κ) det W 1 2 t k i k e ikx iw(k)t+iζ, j where For θ(x, k, t), x = (x 1, x 2, x 3 ) and Eliminating θ, x i t = W(k) k i. ω = θ t, k i = θ x i, ω = W(k, x, t). k i t + ω x i = 0, k i x j k j x i = 0.

Extensions to Higher Dimensions For ω = W(k, x, t) varrying, k i t + W k j k j x i = W x i. k j / x i = k i / x j, k i t + k i x j = W x i, C j (k, x, t) = W k j, C group velocity for some k i, dk i dt = W x i on dx i dt = W k i called the characteristics form. From above, we obtain the Hamiltonian-Jacobi equation θ t θ + W(, x, t) = 0. x

Energy Propagation From α and β constant, ϕ tt α 2 ϕ xx + β 2 ϕ = 0, t (1 2 ϕ2 t + 1 2 α2 ϕ 2 x + 1 2 β2 ϕ 2 ) + x ( α2 ϕ t ϕ x ) = 0, and a slow wavetrain ϕ Re(Ae iθ ) = a cos(θ + η), a = A, η = arga, since Energy density 1 2 ϕ2 t = 1 2 ω2 a 2 sin 2 (θ + η), 1 2 (ω2 + α 2 k 2 )a 2 sin 2 (θ + η) + β 2 a 2 cos 2 (θ + η),

Energy Propagation Flux α 2 ωxa 2 sin 2 (θ + η), with slow varrying ω, k. We obtain E 1 = 1 4 (ω2 + α 2 k 2 + β 2 )a 2, E 2 = 1 2 α2 ωka 2, ω = α 2 k 2 + β 2, E 1 = 1 2 (α2 k 2 + β 2 )a 2, E 2 = 1 2 α2 k α 2 k 2 β 2 a 2 with C(k) and E equation, C(k) = α 2 k α 2 k 2 + β 2, E 2 = C(k)E 1, E 1 t + (CE 1) t gives the differential form of energy. = 0,

Conclusion Basic notions and definition of Linear dispersive waves. Different modes propagate with different wave speeds. General solution and asymptotic analysis. Group and phase velocities. Energy propagation.

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