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

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

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

Wave equation 2D Position of wave front: Wave vector

Harmonic wave Wave 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)

Group velocity Superposition of 2 waves with slightly different k i and  i : Modulated wave Velocity of propogation of modulation = group velocity In the limit of small ,  k with

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. Wave equations which lead to dispersion, have an addtional term: Waves in water aee dispersive. e.g. deep water waves resp. with solution  (kx-  t)

Damped wave Non linear wave equations Wave equations with an addtional time derivative term lead to damped waves with yields resp. With a<2k one obtains 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.

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

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

Capillary waves For water:  = 1000 kg/m 3  = N/m 1 = 1.71 cm c 1 = 23.1 cm/s In addition to pressure force the surface tension is acting as restoring force Bernoulli along pathline Radius of curvature R of water surface for <<  1 for >> 1 Wave length, at which capillary and gravity contributions are equal

Waves at finite water depth

Shallow water equations for h <<  2 In deep water In shallow water hydrostatic pressure distribution

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

Seiches Assumption: Water movement horizontally Linearised equation: Base

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)

Internal waves mit  H -  E Pressure in Epilimnion/equ. of motion: Pressure in Hypolimnion/equ. of motion Continuity: And finally: with

Numerical example Example for basic period of surface seiches und internal seiches –Length of lake: L = 20 km –Depth: Average h = 50 m, epilimnion h E = 10 m, hypolimnion h H = 40 m –Density difference H/E  = Surface-Seiche v = (gh) 1/2 = 22.2 m/s T 1 = 1800 s = 0.5 hours Internal Seiche g‘ = g  = ms -2 h‘ = 8m v = (g‘h‘) 1/2 = 0.28 m/s T 1 = s = 39.7 hours