Waves

1. Waves Numerical Hydraulics W. Kinzelbach M. Wolf C. Beffa

2. Wave equation in 1D u amplitude, v phase velocity k wave number, w angular frequency w = 2pf, f frequency Insertion yields

3. Wave equation 2D Analogous in 2D: 1D-wave front arbitrarily orientied, amplitude u Solution Wave velocity by insertion

4. Wave equation 2D Position of wave front: Wave vector

5. Harmonic waveWave vector in x-direction More economic way of writing Decomposition of an arbitrary wave into harmonic waves (Fourier integral) If domain has finite length L: Only integer k (Fourier analysis)

6. Group velocity Superposition of 2 waves with slightly different ki and wi: with Modulated wave Velocity of propogation of modulation = group velocity In the limit of small Dw, Dk

7. Dispersion If v is constant (independent of k) we get If group and phase velocity are different the wave packet is smoothed out, as the components move with different velocities. This phenomenon is called dispersion. Waves in water aee dispersive. e.g. deep water waves Wave equations which lead to dispersion, have an addtional term: resp. with solution F(kx-wt)

8. Damped wave Wave equations with an addtional time derivative term lead to damped waves with yields resp. With a<2k one obtains Non linear wave equations Non linear wave equations lead to a coupling of harmonic components. There is no more undisturbed superposition but rather interaction (enery exchange) between waves with different k.

9. Types of waves • Gravity waves • are caused by gravity • Capillary waves : • important force is surface tension • Shallow water wave • Gravity wave, but at small water depth (compared to wave length) • Solitons (Surge waves) • Waves with a constant wave profile • Internal waves, seiches 

10. Gravity waves in deep water Path lines of water particles: Circles c wave velocity Bernoulli along water surface l/4 Decrease of amplitude with depth Phase velocity c and group velocity c* of the wave Gravity waves are dispersive

11. Capillary waves In addition to pressure force the surface tension is acting as restoring force Bernoulli along pathline Radius of curvature R of water surface Wave length, at which capillary and gravity contributions are equal for l << l1 For water:r = 1000 kg/m3s = 0.073 N/m l1 = 1.71 cm c1 = 23.1 cm/s for l >> l1

12. Waves at finite water depth

13. Shallow water equations In deep water In shallow water hydrostatic pressure distribution for h << l/2

14. Solitons • Dispersion (small kh) • Front steepening Soliton: Equilibrium between steepening and dispersion mit Wave form does not change

15. Seiches Base Assumption: Water movement horizontally Linearised equation:

16. Surface seiches Assumption: Velocity u constant over depth z Derivative with resp. to t Derivative with resp. to x From those: Standing wave (n-th Oberschwingung)

17. Internal waves Pressure in Epilimnion/equ. of motion: Pressure in Hypolimnion/equ. of motion mit Dr = rH-rE Continuity: And finally: with

18. Numerical example • Example for basic period of surface seiches und internal seiches • Length of lake: L = 20 km • Depth: Average h = 50 m, epilimnion hE = 10 m, hypolimnion hH = 40 m • Density difference H/E Dr/r = 10-3 • Surface-Seiche v = (gh)1/2 = 22.2 m/s T1 = 1800 s = 0.5 hours • Internal Seiche g‘ = gDr/r = 9.81 10-3ms-2 h‘ = 8m v = (g‘h‘)1/2 = 0.28 m/s T1 = 1.43 105 s = 39.7 hours