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Velocity and Pressure in Rear-End Collisions Between Two Solitary Waves With and Without an Underlying Current

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In this paper, we present some theoretical results and we use modern electronic equipment to investigate experimentally rear-end collisions between two solitary waves. The surface position, water-particle velocity, and water pressure of the solitary waves are studied for surface waves in an irrotational water flow without an underlying current, with a (uniform) underlying current flowing in the same direction as the wave (“following” current), and a current flowing in the direction opposite to that of the wave (“opposing” current). The experiments involve a small leading wave that is overtaken by a large trailing wave, with a “compound” wave created when the waves overlap. The wave height, the water-particle velocity, and the water pressure are measured as functions of time using a wave gauge, an electromagnetic meter, and a pressure transducer, respectively. These measurements show that in all three scenarios the compound-wave amplitude decreases during rear-end collisions and reaches a minimum when the wave crests overlap. The measured water-particle velocity of a single solitary wave is also compared with that predicted by second- and third-order solutions of the governing equations, and the measured vertical distribution of pressure is compared with the theoretical vertical distribution of pressure with and without a current. We also study how the current direction (i.e., following or opposing) affects a solitary wave and present extensive results of water velocity and pressure for rear-end collisions between two solitary waves. For a following current, the wave velocity is greater, but the pressure is less than for a wave with a zero current, while for an opposing current the situation is reversed. This indicates that an underlying current affects the water-particle velocity field and that velocity and pressure are related in rear-end collisions. Thus, the water-particle velocity and pressure in the compound wave are greater than in the small wave and less than in the large wave.

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References

  1. Airy, G.B.: Tides and Waves. Encyclopaedia Metropolitana. William Clowes and Sons, London (1841)

    Google Scholar 

  2. Amick, C.J., Toland, J.F.: On periodic water-waves and their convergence to solitary waves in the long-wave limit. Philos. Trans. R. Soc. Lond. Ser. A 303, 633–669 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  3. Boussinesq, J.: Théorie de l’intumescence liquide appeleé onde solitaire ou de translation, se propageant dans un canal rectangulaire. C. R. Acad. Sci. Paris 72, 755–759 (1871)

    MATH  Google Scholar 

  4. Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55–108 (1872)

    MathSciNet  MATH  Google Scholar 

  5. Byatt-Smith, J.G.B.: An integral equation for unsteady surface waves and a comment on the Boussinesq equation. J. Fluid Mech. 49, 625–633 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  6. Byatt-Smith, J.G.B.: On the change of amplitude of interacting solitary waves. J. Fluid Mech. 182, 485–497 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  7. Byatt-Smith, J.G.B.: Perturbation theory for approximately integrable partial differential equations, and the change of amplitude of solitary-wave solutions of the BBM equation. J. Fluid Mech. 182, 467–483 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  8. Chan, R.K.C., Street, R.L.: A computer study of finite-amplitude water waves. J. Comput. Phys. 6, 68–94 (1970)

    Article  ADS  Google Scholar 

  9. Chen, Y., Yeh, H.: Laboratory experiments on counter-propagating collisions of solitary waves, Part 1. Wave interactions. J. Fluid Mech. 749, 577–596 (2014)

    Article  ADS  Google Scholar 

  10. Clamond, D.: Note on the velocity and related fields of steady irrotational two-dimensional surface gravity waves. Philos. Trans. R. Soc. Lond. A 370, 1572–1586 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  11. Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  12. Constantin, A.: On the particle paths in solitary water waves. Q. Appl. Math. 68, 81–90 (2010)

    Article  MathSciNet  Google Scholar 

  13. Constantin, A.: Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, 81, p. 321. SIAM, Philadelphia (2011)

    Book  Google Scholar 

  14. Constantin, A.: Extrema of the dynamic pressure in an irrotational regular wave train. Phys. Fluids 28, Art. 113604 (2016)

    Article  ADS  Google Scholar 

  15. Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431 (2007)

    Article  MathSciNet  Google Scholar 

  16. Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)

    Article  MathSciNet  Google Scholar 

  17. Constantin, A., Escher, J., Hsu, H.-C.: Pressure beneath a solitary water wave: mathematical theory and experiments. Arch. Ration. Mech. Anal. 201, 251–269 (2011)

    Article  MathSciNet  Google Scholar 

  18. Constantin, A., Strauss, W.: Pressure beneath a Stokes wave. Commun. Pure Appl. Math. 53, 533–557 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Craig, W.: Non-existence of solitary water waves in three dimensions. Philos. Trans. R. Soc. Lond. Ser. A 360, 2127–2135 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  20. Craig, W., Guyenne, P., Hammack, J., Henderson, D., Sulem, C.: Solitary water wave interactions. Phys. Fluids 18, 057106 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  21. Craig, W., Sternberg, P.: Symmetry of solitary waves. Commun. Partial Differ. Equ. 13, 603–633 (1988)

    Article  MathSciNet  Google Scholar 

  22. da Silva, A.F.T., Peregrine, D.H.: Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281–302 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  23. Dean, R.G., Dalrymple, R.A.: Water Wave Mechanics for Engineers and Scientists, p. 353. Prentice-Hall, Englewood Cliffs, NJ (1984)

    Google Scholar 

  24. Drazin, P.G., Johnson, R.S.: Solitons: An Introduction, p. 226. Cambridge University Press, Cambridge (1989)

    Book  Google Scholar 

  25. Fenton, J.D.: A ninth-order solution for the solitary wave. J. Fluid Mech. 53, 257–271 (1972)

    Article  ADS  Google Scholar 

  26. Fenton, J.D., Rienecker, M.: Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions. J. Fluid Mech. 118, 411–443 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  27. Friedrichs, K.O., Hyers, D.H.: The existence of solitary waves. Commun. Pure Appl. Math. 7, 517–550 (1954)

    Article  MathSciNet  Google Scholar 

  28. Genoud, F.: Extrema of the dynamic pressure in a solitary wave. Nonlinear Anal. 155, 65–71 (2017)

    Article  MathSciNet  Google Scholar 

  29. Green, G.: On the motion of waves in a variable canal of small depth and width. Trans. Camb. Philos. Soc. 6, 457–462 (1838)

    ADS  Google Scholar 

  30. Green, G.: Note on the motion of waves in canals. Trans. Camb. Philos. Soc. 7, 87–96 (1839)

    ADS  Google Scholar 

  31. Grimshaw, R.: The solitary wave in water of variable depth. J. Fluid Mech. 46, 611–622 (1971)

    Article  ADS  Google Scholar 

  32. Hammack, J., Segur, H.: The Korteweg-de Vries equation and water waves, Part 2. Comparison with experiments. J. Fluid Mech. 65, 289–314 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  33. Henry, D.: On the pressure transfer function for solitary water waves with vorticity. Math. Ann. 357, 23–30 (2013)

    Article  MathSciNet  Google Scholar 

  34. Hur, V.M.: Analyticity of rotational flows beneath solitary water waves. Int. Math. Res. Not. 2012, 2550–2570 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Johnson, R.S.: A Modern Introduction to the Mathematical Theory of Water Waves, p. 445. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  36. Kelland, P.: On the theory of waves, Part 1. Trans. R. Soc. Edinb. 14, 497–545 (1840)

    Article  Google Scholar 

  37. Kelland, P.: On the theory of waves, Part 2. Trans. R. Soc. Edinb. 15, 101–144 (1844)

    Article  Google Scholar 

  38. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal and a new type of long stationary waves. Philos. Mag. 39, 422–443 (1895)

    Article  MathSciNet  Google Scholar 

  39. Laitone, E.V.: The second approximation to cnoidal and solitary waves. J. Fluid Mech. 9, 430–444 (1960)

    Article  ADS  MathSciNet  Google Scholar 

  40. Lewy, H.: A note on harmonic functions and a hydrodynamical application. Proc. Am. Math. Soc. 3, 111–113 (1952)

    Article  MathSciNet  Google Scholar 

  41. Maxworthy, T.: Experiments on collisions between solitary waves. J. Fluid Mech. 76, 177–185 (1976)

    Article  ADS  Google Scholar 

  42. McCowan, J.: On the highest wave of permanent type. Lond. Edinb. Dublin Philos. Mag. 38, 351–358 (1894)

    Article  Google Scholar 

  43. Marchant, T.R., Smyth, N.F.: The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography. J. Fluid Mech. 221, 263–288 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  44. Mirie, R.M., Su, C.H.: Collisions between two solitary waves, Part 2. A numerical study. J. Fluid Mech. 115, 475–492 (1982)

    Article  ADS  Google Scholar 

  45. Rayleigh, B.: On waves. Philos. Mag. 1, 257–279 (1876)

    Article  Google Scholar 

  46. Renouard, D.F., Santos, F., Temperville, A.: Experimental study of the generation, damping, and reflexion of a solitary wave. Dyn. Atmos. Oceans 9, 341–358 (1985)

    Article  ADS  Google Scholar 

  47. Scott Russell, J. Robinson, (Sir) J.: Report on waves. Brit. Assoc. Rep. 417–496 (1837)

  48. Scott Russell, J.: Report on waves. Rep. Meet. Brit. Assoc. Adv. Sci. 14, 311–390 (1844)

    Google Scholar 

  49. Solensen, R.: Basic Wave Mechanics. Wiley, New York (1993)

    Google Scholar 

  50. Stoker, J.J.: Water Waves: The Mathematical Theory with Applications, p. 567. Interscience Publishers Inc, New York (1957)

    MATH  Google Scholar 

  51. Su, C.H., Mirie, R.M.: On head-on collisions between two solitary waves. J. Fluid Mech. 98, 509–525 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  52. Umeyama, M.: Changes in turbulent flow structure under combined wave-current motions. J. Water. Port Coast. Ocean Eng. 135, 213–227 (2009)

    Article  Google Scholar 

  53. Umeyama, M.: Eulerian–Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry. Philos. Trans. R. Soc. Lond. A 370, 1687–1702 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  54. Umeyama, M.: Investigation of single and multiple solitary waves using superresolution PIV. J. Water. Port Coast. Ocean Eng. 139, 303–313 (2013)

    Google Scholar 

  55. Umeyama, M.: Visualization analyses on a head-on collision between two solitary waves. Proc. ICCE ASCE 35, 14 (2016)

    Google Scholar 

  56. Umeyama, M.: Experimental study of head-on and rear-end collisions of two unequal solitary waves. Ocean Eng. 137, 174–192 (2017)

    Article  Google Scholar 

  57. Umeyama, M.: Dynamic-pressure distributions under Stokes waves with and without a current. Philos. Trans. R. Soc. Lond. A 375, Art. 20170103 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  58. Umeyama, M.: Theoretical considerations, flow visualization and pressure measurements for rear-end collisions of two unequal solitary waves. J. Math. Fluid. Mech (2019). https://doi.org/10.1007/s00021-019-0417-6 (to appear)

  59. Umeyama, M., Ishikawa, N., Kobayashi, R.: High-resolution PIV measurements for rear-end and head-on collisions of two solitary waves. Proc. ICCE ASCE 34, 15 (2014)

    Google Scholar 

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  1. Department of Civil and Environmental Engineering, Tokyo Metropolitan University, 1-1 Minamiohsawa Hachioji, Tokyo, 192-0397, Japan

    Motohiko Umeyama

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Umeyama, M. Velocity and Pressure in Rear-End Collisions Between Two Solitary Waves With and Without an Underlying Current. J. Math. Fluid Mech. 21, 37 (2019). https://doi.org/10.1007/s00021-019-0442-5

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