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PHY 7 11 Classical Mechanics and Mathematical Methods 9-9:50 AM MWF Olin 107

This lecture focuses on the analysis of non-linear surface waves in incompressible fluids, including the derivation of soliton solutions. It covers various aspects such as defining equations, boundary conditions, and the integration and rearrangement of coupled equations. The lecture concludes with a summary of the soliton solution.

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PHY 7 11 Classical Mechanics and Mathematical Methods 9-9:50 AM MWF Olin 107

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  1. PHY 711 Classical Mechanics and Mathematical Methods • 9-9:50 AM MWF Olin 107 • Plan for Lecture 33: • Chapter 10 in F & W: Surface waves • -- Non-linear contributions and soliton solutions PHY 711 Fall 2017 -- Lecture 33

  2. W PHY 711 Fall 2017 -- Lecture 33

  3. Surface waves in an incompressible fluid • General problem • including • non-linearities p0 z z h y x PHY 711 Fall 2017 -- Lecture 33

  4. p0 z z h y x PHY 711 Fall 2017 -- Lecture 33

  5. Non-linear effects in surface waves: z=0 p0 z z h y x PHY 711 Fall 2017 -- Lecture 33

  6. Detailed analysis of non-linear surface waves [Note that these derivations follow Alexander L. Fetter and John Dirk Walecka, Theoretical Mechanics of Particles and Continua(McGraw Hill, 1980), Chapt. 10.] The surface of the fluid is described by z=h+z(x,t). It is assumed that the fluid is contained in a structure (lake, river, swimming pool, etc.) with a structureless bottom defined by the z = 0 plane and filled to an equilibrium height of z = h. PHY 711 Fall 2017 -- Lecture 33

  7. Defining equations for F(x,z,t)and z(x,t) Bernoulli equation (assuming irrotational flow) and gravitation potential energy PHY 711 Fall 2017 -- Lecture 33

  8. Boundary conditions on functions – Zero velocity at bottom of tank: Consistent vertical velocity at water surface PHY 711 Fall 2017 -- Lecture 33

  9. Analysis assuming water height z is small relative tovariations in the direction of wave motion (x)Taylor’s expansion about z = 0: Note that the zero vertical velocity at the bottom ensures that all odd derivatives vanish from the Taylor expansion . In addition, the Laplace equation allows us to convert all even derivatives with respect to zto derivatives with respect to x. PHY 711 Fall 2017 -- Lecture 33

  10. Check linearized equations and their solutions: Bernoulli equations -- Using Taylor's expansion results to lowest order PHY 711 Fall 2017 -- Lecture 33

  11. Analysis of non-linear equations -- keeping the lowest order nonlinear terms and include up to 4th order derivatives in the linear terms. The expressions keep the lowest order nonlinear terms and include up to 4th order derivatives in the linear terms. PHY 711 Fall 2017 -- Lecture 33

  12. Note that the wave “speed” cwill be consistently determined PHY 711 Fall 2017 -- Lecture 33

  13. Integrating and re-arranging coupled equations PHY 711 Fall 2017 -- Lecture 33

  14. Integrating and re-arranging coupled equations – continued -- Expressing modified surface velocity equation in terms of h(u): PHY 711 Fall 2017 -- Lecture 33

  15. Solution of the famous Korteweg-de Vriesequation Modified surface amplitude equation in terms of h Soliton solution PHY 711 Fall 2017 -- Lecture 33

  16. PHY 711 Fall 2017 -- Lecture 33

  17. Relationship to “standard” form of Korteweg-de Vries equation PHY 711 Fall 2017 -- Lecture 33

  18. More details PHY 711 Fall 2017 -- Lecture 33

  19. PHY 711 Fall 2017 -- Lecture 33

  20. Summary Soliton solution PHY 711 Fall 2017 -- Lecture 33

  21. Photo of canal solitonhttp://www.ma.hw.ac.uk/solitons/ Some links: Website – http://www.ma.hw.ac.uk/solitons/ PHY 711 Fall 2017 -- Lecture 33

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