T. Brooke Benjamin ... a brilliant researcher .... original and elegant applied mathematics ... extraordinary physical insights ... changed the way we now think about hydrodynamics ... about 6ft tall ... moved slowly ... smiled slightly ... often smoking a pipe ... every word counted ... ... he and his wife Natalia were a joyous couple with many friends ... Mathematical Aspects of Classical Water Wave Theory from the Past 20 Year - the fascination of what’s difficult - Brooke Benjamin Lecture Oxford 26th November 2013 Classical Water Waves What distinguishes the water wave problems from others in hydrodynamics is that the unknown is the region occupied by the fluid. It is referred to as the free surface Classical Water Waves What distinguishes the water wave problems from others in hydrodynamics is that the unknown is the region occupied by the fluid. It is referred to as the free surface For today Classical means two-dimensional irrotational flow under gravity of a heavy perfect liquid with a free surface, Classical Water Waves What distinguishes the water wave problems from others in hydrodynamics is that the unknown is the region occupied by the fluid. It is referred to as the free surface For today Classical means two-dimensional irrotational flow under gravity of a heavy perfect liquid with a free surface, There is no surface tension and there are no floating bodies Classical Water Waves What distinguishes the water wave problems from others in hydrodynamics is that the unknown is the region occupied by the fluid. It is referred to as the free surface For today Classical means two-dimensional irrotational flow under gravity of a heavy perfect liquid with a free surface, There is no surface tension and there are no floating bodies Although there has been important recent progress on the initial-value problem for free boundaries and on fully 3-dimensional waves, that is not the concern here. Classical Water Waves What distinguishes the water wave problems from others in hydrodynamics is that the unknown is the region occupied by the fluid. It is referred to as the free surface For today Classical means two-dimensional irrotational flow under gravity of a heavy perfect liquid with a free surface, There is no surface tension and there are no floating bodies Although there has been important recent progress on the initial-value problem for free boundaries and on fully 3-dimensional waves, that is not the concern here. 2-Dimensional Irrotational Water Waves: Classical Water Waves What distinguishes the water wave problems from others in hydrodynamics is that the unknown is the region occupied by the fluid. It is referred to as the free surface For today Classical means two-dimensional irrotational flow under gravity of a heavy perfect liquid with a free surface, There is no surface tension and there are no floating bodies Although there has been important recent progress on the initial-value problem for free boundaries and on fully 3-dimensional waves, that is not the concern here. 2-Dimensional Irrotational Water Waves: In Eulerian co ordinates the velocity at a point (x, y, z) in the fluid at time t is given by the gradient of a scalar potential φ on R2 v~ (x, y, z; t)= ∇φ(x, y; t) which satisfy Irrotational Water Waves Infinite depth Wave interior Ω= {(x, y) : y < η(x, t)} ∆φ(x, y; t)= 0 ∇φ(x, y; t) → 0 as y → −∞ Irrotational Water Waves Boundary Conditions gravity g acts vertically down Wave Surface S = {(x, η(x, t)) : x ∈ R} φ + 1 |∇φ|2 + gy = 0 t 2 on S η + φ η − φ = 0 t x x y This is a very old problem, studied by Cauchy, Laplace, Lagrange, Poisson, Green, Airy, Stokes, Rayleigh ... This is a very old problem, studied by Cauchy, Laplace, Lagrange, Poisson, Green, Airy, Stokes, Rayleigh ... After he discovered experimentally the solitary wave, in 1844 John Scott Russell described This is a very old problem, studied by Cauchy, Laplace, Lagrange, Poisson, Green, Airy, Stokes, Rayleigh ... After he discovered experimentally the solitary wave, in 1844 John Scott Russell described the greater part of the investigations of Poisson and Cauchy under the name of wave theory are rather to be regarded as mathematical exercises, not physical investigations This is a very old problem, studied by Cauchy, Laplace, Lagrange, Poisson, Green, Airy, Stokes, Rayleigh ... After he discovered experimentally the solitary wave, in 1844 John Scott Russell described the greater part of the investigations of Poisson and Cauchy under the name of wave theory are rather to be regarded as mathematical exercises, not physical investigations ... since no one had predicted the wave it remained for mathematicians to give an “a priori demonstration a posteriori” This is a very old problem, studied by Cauchy, Laplace, Lagrange, Poisson, Green, Airy, Stokes, Rayleigh ... After he discovered experimentally the solitary wave, in 1844 John Scott Russell described the greater part of the investigations of Poisson and Cauchy under the name of wave theory are rather to be regarded as mathematical exercises, not physical investigations ... since no one had predicted the wave it remained for mathematicians to give an “a priori demonstration a posteriori” Progress was very slow throughout the 19th century with approximation for certain scaling limits but the full equations remained essentially unexamined for more than one hundred years but progress had accelerated hugely in recent years This is a very old problem, studied by Cauchy, Laplace, Lagrange, Poisson, Green, Airy, Stokes, Rayleigh ... After he discovered experimentally the solitary wave, in 1844 John Scott Russell described the greater part of the investigations of Poisson and Cauchy under the name of wave theory are rather to be regarded as mathematical exercises, not physical investigations ... since no one had predicted the wave it remained for mathematicians to give an “a priori demonstration a posteriori” Progress was very slow throughout the 19th century with approximation for certain scaling limits but the full equations remained essentially unexamined for more than one hundred years but progress had accelerated hugely in recent years Alex Craik, a PhD student of Brooke’s, has written extensively on the history of the water wave problem Energetics The total wave energy at time t is Kinetic + Potential: 1 η(x,t) g |∇φ(x, y; t)dy|2dx + η2(x; t)dx 2 −∞ 2 Z Z Z Energetics The total wave energy at time t is Kinetic + Potential: 1 η(x,t) g |∇φ(x, y; t)dy|2dx + η2(x; t)dx 2 −∞ 2 Z Z Z Given periodic functions η and Φ of the single variable x let Ω= {(x, y): y < η(x)} Energetics The total wave energy at time t is Kinetic + Potential: 1 η(x,t) g |∇φ(x, y; t)dy|2dx + η2(x; t)dx 2 −∞ 2 Z Z Z Given periodic functions η and Φ of the single variable x let Ω= {(x, y): y < η(x)} φ the solution of the corresponding Dirichlet problem ∆φ(x, y)= 0 on Ω φ → 0 as y → −∞ φ(x, η(x)) = Φ(x) Energetics The total wave energy at time t is Kinetic + Potential: 1 η(x,t) g |∇φ(x, y; t)dy|2dx + η2(x; t)dx 2 −∞ 2 Z Z Z Given periodic functions η and Φ of the single variable x let Ω= {(x, y): y < η(x)} φ the solution of the corresponding Dirichlet problem ∆φ(x, y)= 0 on Ω φ → 0 as y → −∞ φ(x, η(x)) = Φ(x) let 1 η(x) E(η, Φ) = |∇φ(x, y)dy|2dx + gη2(x)dx 2 −∞ Z Z With this functional 1 η(x) E(η, Φ) := |∇φ(x, y)dy|2dx + gη2(x)dx 2 −∞ Z Z and with the ”variational” derivatives ∂E ∂E and ∂Φ ∂η With this functional 1 η(x) E(η, Φ) := |∇φ(x, y)dy|2dx + gη2(x)dx 2 −∞ Z Z and with the ”variational” derivatives ∂E ∂E and ∂Φ ∂η Zakharov (1968) observed that solutions (η, Φ) of ∂η ∂E ∂Φ ∂E = (η, Φ); = − (η, Φ) ∂t ∂Φ ∂t ∂η yields a water wave Benjamin & Olver (1982) studied ∂η ∂E ∂Φ ∂E = (η, Φ); = − (η, Φ) ∂t ∂Φ ∂t ∂η as a Hamiltonian system of classical type 0, I x˙ = J∇E(x), x = (η, Φ), J = , −I, 0 η, φ being the infinite dimensional canonical variables which they referred to as ”coordinates” and ”momentum” and in an Appendix gives the Hamiltonian formulation independent of coordinates. Both Zakharov and Benjamin was conscious of the implications of the Hamiltonian formulation for stability Spatially Periodic Waves Normalised Period 2π Spatially Periodic Waves Normalised Period 2π Based on conformal mapping theory any rectifiable periodic Jordan curve S = {(x, η(x)) : x ∈ R} can be re-parametrised as S = {(−ξ −Cw(ξ), w(ξ)) : ξ ∈ R} where Cw is the Hilbert transform of a periodic function w: 1 π w(σ) dσ Cw(ξ) = pv 2 Z−π tan 1 − π 2 (ξ σ) Spatially Periodic Waves Normalised Period 2π Based on conformal mapping theory any rectifiable periodic Jordan curve S = {(x, η(x)) : x ∈ R} can be re-parametrised as S = {(−ξ −Cw(ξ), w(ξ)) : ξ ∈ R} where Cw is the Hilbert transform of a periodic function w: 1 π w(σ) dσ Cw(ξ) = pv 2 Z−π tan 1 − π 2 (ξ σ) Dyachenko, Kuznetsov, Spector & Zakharov (1996) used complex function theory in a deep and beautiful way to reduce Zakharov’s awkward system to the following ”simple” system: 0 0 0 w˙ (1 + Cw ) −Cϕ − w Cw˙ = 0 0 0 0 0 0 C w ϕ˙ − w˙ ϕ + λww + (ϕ ˙ + λw)(1 + Cw ) − ϕ Cw˙ = 0 0 ˙ =∂/∂t, = ∂/∂x w = wave height ϕ = potential at surface: 0 < λ = gravity after normalisating the wavelength as 2π Spatially Periodic Waves Normalised Period 2π Based on conformal mapping theory any rectifiable periodic Jordan curve S = {(x, η(x)) : x ∈ R} can be re-parametrised as S = {(−ξ −Cw(ξ), w(ξ)) : ξ ∈ R} where Cw is the Hilbert transform of a periodic function w: 1 π w(σ) dσ Cw(ξ) = pv 2 Z−π tan 1 − π 2 (ξ σ) Dyachenko, Kuznetsov, Spector & Zakharov (1996) used complex function theory in a deep and beautiful way to reduce Zakharov’s awkward