Mod elisation math ematique des vagues David Lannes Institut de - PowerPoint PPT Presentation

Mod´ elisation math´ ematique des vagues David Lannes Institut de Math´ ematiques de Bordeaux et CNRS UMR 5251 Journ´ ee des doctorants David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 1 / 30

Goal David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 2 / 30

Goal David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 2 / 30

Goal David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 2 / 30

Where do waves come from? How are they created? Source: Les vagues en ´ equations, Pour la Science, no 409, novembre 2011 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 3 / 30

Where do waves come from? What is their speed? Sir Isaac Newton (1642-1727) Principia Mathematica , 1687 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 4 / 30

Where do waves come from? What is their speed? David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 4 / 30

Where do waves come from? What is their speed? Leonhard Euler (1707-1783) M´ emoires de l’Acad´ emie royale des sciences et des belles lettres de Berlin , 1757 � Equations of fluid mechanics ρ ( ∂ t U + U · ∇ X , z U ) = − ∇ X , z P + ρ g div U =0 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 5 / 30

Where do waves come from? What is their speed? Leonhard Euler (1707-1783) M´ emoires de l’Acad´ emie royale des sciences et des belles lettres de Berlin , 1757 � Equations of fluid mechanics ρ ( ∂ t U + U · ∇ X , z U ) = − ∇ X , z P + ρ g div U =0 This equations are very general What do they tell us about waves? David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 5 / 30

Where do waves come from? What is their speed? Giuseppe Lodovico Lagrangia (Joseph Louis Lagrange) (1736-1813) M´ emoire sur la th´ eorie du mouvement des fluides , 1781 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 6 / 30

Where do waves come from? What is their speed? David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 6 / 30

Where do waves come from? What is their speed? Source: Les vagues en ´ equations, Pour la Science, no 409, novembre 2011 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 7 / 30

Where do waves come from? What is their speed? Comparison of Newton and Lagrange’s formulas Lagrange: c = √ gH . � All waves have same speed David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 8 / 30

Where do waves come from? What is their speed? Comparison of Newton and Lagrange’s formulas Lagrange: c = √ gH . � All waves have same speed √ gL where L is the wave length of the wave 1 Newton: c = √ 2 π � Waves of different wavelength propagate differently David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 8 / 30

Where do waves come from? What is their speed? Comparison of Newton and Lagrange’s formulas Lagrange: c = √ gH . � All waves have same speed √ gL where L is the wave length of the wave 1 Newton: c = √ 2 π � Waves of different wavelength propagate differently This is dispersion: Source: Les vagues en ´ equations, Pour la Science, no 409, novembre 2011 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 8 / 30

Where do waves come from? What is their speed? Comparison of Newton and Lagrange’s formulas Lagrange: c = √ gH . � All waves have same speed √ gL where L is the wave length of the wave 1 Newton: c = √ 2 π � Waves of different wavelength propagate differently Comparison for a wave a 0 ( x ) = sin( x ) + 0 . 5 sin(2 x ). Source: Les vagues en ´ equations, Pour la Science, no 409, novembre 2011 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 8 / 30

Where do waves come from? What is their speed? Recall how waves are created Source: Les vagues en ´ equations, Pour la Science, no 409, novembre 2011 So the good formula should be David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 9 / 30

Where do waves come from? What is their speed? Recall how waves are created Source: Les vagues en ´ equations, Pour la Science, no 409, novembre 2011 So the good formula should be √ gL 1 Newton: c = √ 2 π David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 9 / 30

Closer to the shore Another formula! Closer to the shore we observe: And the relevant formula is David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 10 / 30

Closer to the shore Another formula! Closer to the shore we observe: And the relevant formula is Lagrange: c = √ gH David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 10 / 30

Closer to the shore What happens? Sim´ eon Denis Augustin Louis Sir George Biddell Sir George Gabriel Poisson Cauchy Airy Stokes (1789–1857) (1801–1892) (1819–1903) (1780–1840) David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 11 / 30

Closer to the shore What happens? Sim´ eon Denis Augustin Louis Sir George Biddell Sir George Gabriel Poisson Cauchy Airy Stokes (1789–1857) (1801–1892) (1819–1903) (1780–1840) � A single formula with two different asymptotic regimes Lagrange’s formula in shallow water ( H / L → 0), Newton’s formula in deep water ( H / L → ∞ ). David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 11 / 30

Modern mathematical approaches Notations David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 12 / 30

Modern mathematical approaches The free surface Euler equations The free surface Euler equations 1 ∂ t U + ( U · ∇ X , z ) U = − 1 ρ ∇ X , z P − g e z in Ω t David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

Modern mathematical approaches The free surface Euler equations The free surface Euler equations 1 ∂ t U + ( U · ∇ X , z ) U = − 1 ρ ∇ X , z P − g e z in Ω t 2 div U = 0 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

Modern mathematical approaches The free surface Euler equations The free surface Euler equations 1 ∂ t U + ( U · ∇ X , z ) U = − 1 ρ ∇ X , z P − g e z in Ω t 2 div U = 0 3 curl U = 0 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

Modern mathematical approaches The free surface Euler equations The free surface Euler equations 1 ∂ t U + ( U · ∇ X , z ) U = − 1 ρ ∇ X , z P − g e z in Ω t 2 div U = 0 3 curl U = 0 4 Ω t = { ( X , z ) ∈ R d +1 , − H 0 + b ( X ) < z < ζ ( t , X ) } . David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

Modern mathematical approaches The free surface Euler equations The free surface Euler equations 1 ∂ t U + ( U · ∇ X , z ) U = − 1 ρ ∇ X , z P − g e z in Ω t 2 div U = 0 3 curl U = 0 4 Ω t = { ( X , z ) ∈ R d +1 , − H 0 + b ( X ) < z < ζ ( t , X ) } . 5 U · n = 0 on { z = − H 0 + b ( X ) } . David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

Modern mathematical approaches The free surface Euler equations The free surface Euler equations 1 ∂ t U + ( U · ∇ X , z ) U = − 1 ρ ∇ X , z P − g e z in Ω t 2 div U = 0 3 curl U = 0 4 Ω t = { ( X , z ) ∈ R d +1 , − H 0 + b ( X ) < z < ζ ( t , X ) } . 5 U · n = 0 on { z = − H 0 + b ( X ) } . � 6 ∂ t ζ − 1 + |∇ ζ | 2 U · n = 0 on { z = ζ ( t , X ) } . David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

Modern mathematical approaches The free surface Euler equations The free surface Euler equations 1 ∂ t U + ( U · ∇ X , z ) U = − 1 ρ ∇ X , z P − g e z in Ω t 2 div U = 0 3 curl U = 0 4 Ω t = { ( X , z ) ∈ R d +1 , − H 0 + b ( X ) < z < ζ ( t , X ) } . 5 U · n = 0 on { z = − H 0 + b ( X ) } . � 6 ∂ t ζ − 1 + |∇ ζ | 2 U · n = 0 on { z = ζ ( t , X ) } . 7 P = P atm on { z = ζ ( t , X ) } . David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

Modern mathematical approaches The free surface Euler equations The free surface Euler equations 1 ∂ t U + ( U · ∇ X , z ) U = − 1 ρ ∇ X , z P − g e z in Ω t 2 div U = 0 3 curl U = 0 4 Ω t = { ( X , z ) ∈ R d +1 , − H 0 + b ( X ) < z < ζ ( t , X ) } . 5 U · n = 0 on { z = − H 0 + b ( X ) } . � 6 ∂ t ζ − 1 + |∇ ζ | 2 U · n = 0 on { z = ζ ( t , X ) } . 7 P = P atm on { z = ζ ( t , X ) } . Definition Equations (H1)-(H9) are called free surface Euler equations. David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30