Gravity Waves in Water

Transcription

1 Hydrodynamics Seminar Gravity Waves in Water Benjamin Kellers

2 Agenda Introduction A Brief History of Wave Mechanics Deriving the Velocity of the Wave Dispersion Summary Benjamin Kellers 2

3 Introduction Gravity Waves Water waves Main causing force: gravity Other forces negligibly small (e.g. surface tension) Waves do not transport matter! Only energy! (particle paths are closed elipses) Benjamin Kellers 3

4 Agenda Introduction A Brief History of Wave Mechanics Deriving the Velocity of the Wave Dispersion Summary Benjamin Kellers 4

5 A Brief History of Wave Mechanics Benjamin Kellers 5

6 A Brief History of Wave Mechanics Reproduced from Newton: Principia, Book II, Section VIII Benjamin Kellers 6

7 A Brief History of Wave Mechanics The frequency of deep-water waves is inversely proportional to the square root of the "breadth of the wave." "These things are true upon the supposition that the parts of water ascend or descend in a right line; but, in truth, that ascent and descent is rather performed in a circle; and therefore I propose the time defined by this Proposition as only near the truth." Benjamin Kellers 7

8 A Brief History of Wave Mechanics 1687 Newton, Principia 1757/61 Leonhard Euler, Euler s Equations 1776 Pierre-Simon Laplace, unrecognized progress Benjamin Kellers 8

9 A Brief History of Wave Mechanics 1781/86 Joseph-Louis Lagrange "the speed of propagation of waves will be that which a heavy body would acquire in falling from... half the height of the water in the canal" shallow water approximation! Benjamin Kellers 9

10 A Brief History of Wave Mechanics ca George Biddell Airy and George Gabriel Stokes, first modern derivation of the phase velocity Reproduced from Airy: Tides and Waves (1841/45), p Frequency n, wavenumer m, depth k Benjamin Kellers 10

11 Agenda Introduction A Brief History of Wave Mechanics Deriving the Velocity of the Wave Dispersion Summary Benjamin Kellers 11

12 Deriving the Velocity of the Wave Steps 1. Fundamental equations & boundary cond. 2. Solving this system of equations 3. Expanding the result to discuss special cases Benjamin Kellers 12

13 Agenda Introduction A Brief History of Wave Mechanics Deriving the Velocity of the Wave Dispersion Summary Benjamin Kellers 13

14 Dispersion Questions to be answered: 1. What is the group velocity? 2. How to classify dispersion? 3. What kind of dispersion have our special cases? Benjamin Kellers 14

15 Dispersion Green dots: group velocity Red dots: phase velocity Reproduced from released under CC license, author: Kraaiennest Benjamin Kellers 15

16 Agenda Introduction A Brief History of Wave Mechanics Deriving the Velocity of the Wave Dispersion Summary Benjamin Kellers 16

17 Summary GENERAL CASE Dispersion relation: Phase velocity: Benjamin Kellers 17

18 Summary DEEP-WATER WAVES APPROXIMATION Dispersion relation: Phase velocity: normal dispersion Benjamin Kellers 18

19 Summary SHALLOW-WATER WAVES APPROXIMATION Dispersion relation: Phase velocity: no dispersion Benjamin Kellers 19

20 References Craik, A. D. D.: The Origins of Water Wave Theory, Ann. Rev. Fluid Mech. 36, 1: 1 28 (2004). Newton, I.: The Mathematical Principles of Natural Philosophy, or. publ. by The Royal Society, London (1687). Translated into English by Andrew Motte (1729). accessed: Lagrange, J.-L.: Sur la manière de rectifier deux entroits des Principes de Newton relatifs à la propagation du son et au mouvement des ondes, or. Publ Nouv. Mém. Acad. Berlin (1889). Bestehorn, M.: Hydrodynamik und Strukturbildung, Springer-Verlag Berlin Heidelberg (2006). Airy, G. B.: Tides and Waves: Extracted from the Encyclopaedia Metropolitana, Tom. V Pag , ed. William Clowes and Sons (1845). Stokes, G. G.: On the Theory of Oscillatory Waves, Trans. Cambridge Phil. Soc. 8, (1847). Craik, A. D. D.: George Gabriel Stokes on Water Wave Theory, Ann. Rev. Fluid Mech. 37, 1: (2005). Wolschin, G.: Hydrodynamik, Springer-Verlag Berlin Heidelberg (2016) Benjamin Kellers 20