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Measurements of Resonant Frequencies,Phase Velocities,Q factors, and Damping Coefficients,Alpha, of Dispersive Water Waves and Observations of Solitons.

Dispersive water waves can be seen of surfaces of water due to the inclusion of gravity and surface tension terms in the dispersion relation. These can be created in the lab using a container filled with water to a height,h, and mixed with some salt and fotoflow. Using a function generator and and amplifier sweeps from 0.5-10 Hz and made and observed resonant frequencies are found. Further investigation of the data, one can find the phase velocities of these dispersive waves and q factors. Then by changing the apparatus around, solitions can be made and observed by driving the container at resonant frequencies and adding pertubations to the surface can induce a self induced wave packet.

At low frequencies water is an incompressible fluid and the density is constant. At the surface the pressure oscillation is \(\delta p=0\) due to a plane wave not being able to propogate because of boundary conditions. When factoring gravity in then there is a pressure oscillation that culminates in a bulge in the water. Now \(\delta p=\rho g \delta h\), where \(\rho\) is the liquid density and \(\delta h\) is the height of the wave. Mass conversation is described by \(\nabla ^2 \phi=0\) where \(\phi\) is the velocity potential. Using eulers equation it is found the the pressure oscillation becomes \(\delta p=-\frac{\partial \phi}{\partial t} \rho\). We can look at the boundary conditions for this lab are at z=0,h the velocity of the fluid is \(v_z=0\) at z=0 and \(\delta p=\rho g \delta h\) at z=h.The velocity potential takes on the form \[\phi (x,z,t)=(A\cosh kz+ b\sinh kz)e^{i(kz-\omega t)}\] After applying the boundary conditions listed above, the dispersion relation for these waves are shown to be \[\omega ^2=gk\tanh(kh)\] where k is the wavenumber=\(\frac{2\pi}{\lambda}\), h is the height of the wave and \(\omega\) is the angular frequency. For a shallow water wave make the assumption \(kh<<1\), then recall that the phase velocity is \(v_{phase}=\frac{\omega}{k}\) and in the shallow water limit \(v_{phase}=\sqrt{gh}\). For the general dispersion relation, the phase velocity is \[v_{phase}=\sqrt{\frac{g\tanh kh}{k}}\] At higher frequencies the wavelength becomes comparable the length of the water so include the surface tension in the dispersion relation as \[\omega^2=k(g+\frac{\delta}{\rho} k^2)\tanh(kh)\]

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