\documentclass{article} \begin{document} \pagestyle{empty} {\small \begin{center} Quick and Dirty Wave Simulation \end{center} On the previous web page are links to figures based on a naive simulation of water wave motion. The simulation uses the deep-water approximation in which water particles travel in circles whose heights fall off exponentially with depth below the still-water surface. The wavelength used here is $\lambda$ = 10 meters, which gives a wave velocity of $C = \sqrt{g*\lambda/(2\pi)}$ = 3.9 m/s. The period $T = \lambda/C$ = 2.5 seconds. The amplitude $a$ of the surface waves is set to be 0.5 m, giving a crest/trough height difference of 1.0 m. The resulting speed of individual water particles at the surface is $v = \pi a / T$ = 0.62 m/s. The wave moves toward the right in the picture. For a nominal depth $z$ below the still-water surface, a water molecule's amplitude of orbit is $a e^{-2\pi d/\lambda}$. Figure 1 shows a profile of sample water molecules at $t=0$ and $t= 0.25 T$. Particles of the same color lie in the same vertical column when at the top or bottom of their orbits. The numerical label on each particle gives the depth below surface (arbitrary units) of the center of its orbit. Figure 2 shows the same wave at $t=0.50 T$ and $t=0.75T$. The previous page also contains a link to to access the fortran program used to generate the wave simulation. The program creates an output file containing commands to be read by CERN's PAW graphics display program. } \end{document}