Advanced Water Physics

Moving water

Describing ocean waves is a huge challenge. It has several different components and their cooperation results a very complex system. Basically there are two different kinds of mechanical wave motion: longitudinal and transverse. The direction of oscillation relative to the wave motion distinguishes them. If the oscillation is parallel to the wave motion it is called longitudinal wave:

longitudinal_wave.gif

If the oscillation is perpendicular to the wave motion it is called transversal wave:

transverse_wave.gif

Examples are, for example, sound wave in air for longitudinal waves and the motion of a guitar-string if you play the guitar for transverse waves. For both kinds of waves amplitude is the maximum displacement of a wave from the equilibrium and wavelength is the shortest length between two points of the wave which are in the same wave-phase. The frequency shows the number of wave cycles in a second. It is easy to calculate the speed of a wave from these data: wave speed equals the product of frequency and wavelength.

The blowing wind and gravity together forms the ocean waves which propagate along the surface of the water and air. This compound system has both longitudinal and transverse components and makes water particles move in a circular path. The closer to the surface a particle is, the bigger the radius of the motions becomes. This kind of wave is called surface wave and is illustrated on the next figure:

surface_wave.png

At point A, where the water is deeper, the path is circle, at point B, where the water is shallow, the path of the motion becomes elliptic with decreasing water depth. The arrow #1 shows the direction of propagation. #2 shows the crest of the wave, #3 shows the wave trough.

The next animation tries to visualize the process:

surface_wave_animation.gif

The source of the image is the homepage of Kettering University.

The following equation describes the dispersion relationship of the surface waves (the source is the scienceworld homepage):

surface_wave_equation.gif

where ω is the angular frequency, g is the gravitational acceleration, k is the wavenumber, γ is the surface tension, and ρ is the density. Solving for ω gives

simg799.gif

Compound systems – summation of different waves

The previously discussed theoretical background is enough to describe the motion of ocean waves, but we need to use more components with different amplitudes and wavelengths to get a more realistic result:

more_component_wave.jpg

The sum of the components results the next wave:

more_component_wave_result.jpg

In the tree-dimensional world different components have not only different amplitudes and wavelength, but different directions as well. The dominant direction will be the one with bigger amplitude and longer period. The next figure shows three components:

more_component_3d_wave.jpg

And the sum of the components can approximate the real ocean surface realistic:

more_component_3d_wave_result.jpg

The source of the pictures is http://www.glenbrook.k12.il.us/gbssci/phys/Class/waves/u10l1c.html and http://www.carbontrust.co.uk/technology/technologyaccelerator/ME_guide2.htm

Useful link:

http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16/chapter16_01.htm


Gerstner Waves

A Czech scientist, Jozef Gerstner obtained the first exact solution describing water waves of arbitrary amplitudes in 1802. His model also describes the cycloidal movement of the surface waves. In this model the water depth is large compared to the wave length. The resulted curve is also called trachoid.

Displacements are defined with the following equations:

x = X0-(k/k0) * A sin( k * X0 - ωt)
y = A cos( k * X0 - ωt)

where X0 is the undisturbed surface point, A is the wave amplitude, k is the wave vector and k0 is the magnitude.

Gerstner waves become close to sinusoidal if the amplitudes are very small, but they break if the amplitudes are bigger:

gertner_waves.jpg

These qualities allow Gerstner waves to describe various surface waves under different conditions.

For more details, see [1] or [2].


The Navier-Stokes Equations

Navier-Stokes Equations (NSE) are nonlinear partial differential equations and describe the motion of incompressible viscose fluids. In NSE there are three types of forces acting:

  • Gravity: Fg = ρG, where ρ is the density and G is the gravitational force (9.81 m/s2).
  • Pressure forces: These forces act inwards and normal to the water surface.
  • Viscose forces: These are forces due to friction in the water and acts in all directions on all elements of the water.

The time dependent chaotic, stochastic behavior of fluids is called turbulence. Navier-Stokes equations are thought to describe the phenomena, but it is not answered yet, how to decide whether smooth, physically reasonable solutions exist for the equations. Actually a $1,000,000 prize is offered to whoever makes preliminary progress toward a mathematical theory which will help in the understanding of this phenomenon. Further discussion is outside the scope of this paper. The Navier-Stokes equations are given by:

navier-stokes-equations.jpg?w=544

Where, uº(x) is a given, C divergence-free vector field on Rn, fi(x, t) are the components of a given, externally applied force (e.g. gravity), ν is a positive coefficient (the viscosity), and

navier-stokes-details.jpg

is the Laplacian in the space variables. For more details, see [3], [5] or read about an implemented version: [4].


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