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Extra resources for Aventures mathématiques
5) where the linear partial differential operator L is related to the dispersion function D(k, ω) by the usual Fourier analysis correspondence: L(∂x , ∂t ) ≡ D(−i∂x , i∂t ). 6) The Green function solution is given by G(x, t) = L dω 2π F dk exp[i(kx − ωt)] . 7), the contour F must lie in a strip of the complex k-plane, including the Re(k)-axis (this allows the x > 0 eigenmodes to be separated off from the x < 0 eigenmodes). The integrand will be analytic on F for an appropriate choice of the contour L in the complex ω-plane.
47) for ξ ≤ 0 (solid lines) and ξ > 0 (dashed lines). for the derivatives of ρ. Thus the derivatives of ρ diverge when 1 + tF (ξ) = 0 or t = − 1 . 49). 49) for the derivatives of ρ to diverge. 51) where |F (ξ)| is maximal at ξ = ξB . From a physical standpoint, the wave breaks because the points on the wave profile ρ(x, t) with the larger values of ρ travel at a faster speed than the points with smaller ρ (this assumes λ (ρ) > 0 and that λ(ρ) > 0). 1. At times t > tB , the solution becomes multi-valued, and it is then necessary to insert a shock in the flow in order to obtain a consistent weak solution of the equation.
Hamiltonian description of shear flows is given by Balmforth and Morrison . 3 Wave Breaking and Gradient Catastrophe Wave breaking of nonlinear hyperbolic waves in one Cartesian space dimension can be demonstrated by considering the conservation equation: ρt + [Q(ρ)]x = 0. 41) are detailed in [198, Ch. g. flood waves, traffic flow and simple sound waves 30 1. OVERVIEW OF PARTIAL DIFFERENTIAL EQUATIONS in gas dynamics). The quantity ρ is a density and Q(ρ) is the flux. 43) is the characteristic speed of the wave.