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 \[\omega^2=k(g+\frac{\delta}{\rho} k^2)\tanh(kh)\]

Water waves were observed in a container that had water up to a height of \(0.03\) meters and the length of the container was \(.38\) meters long. A function generator that is connected to an amplifier that drives a speaker that is connected to the glass container at one end. The container is on a bearing stage that allows the container to move with the frequency that is driving the speaker. The movement will cause the surface to oscillate with the oscillator and if a resonant mode is reached, a maximum on the surface oscillation will be seen and this is detected by two wires on the other side of the container that conduct due to the salt in the water. The resistance between the two wires will oscillate with the oscillation in the height of the water. This is then connected to a wheatstone bridge which is a circuit used to find an unknown resistance by balancing two legs of a bridge circuit. Of the four resistances in the wheatstone bridge one is unknown and one of the known resistances is variable. If the ratio of the resistances in the known leg is equal to the ratio of the resistances in the unknown leg, then no current will flow through the galvanometer. If theres an inbalance than you can vary the variable resistance until no current flows. A lock in amplifier is used to measure the voltage from the equal ratios of resistance. This voltage is the reference voltage and the two voltages from the waves themselves are used at A and -B as shown in the diagram below. after filtering the output and converting AC to DC,a DC rms amplitude that is equivalent to the waves amplitude is fed into LabView to be recorded. Data was taken this way for sweeps from 0.5-3.5 Hz,3.5-7 Hz, and 7-10 Hz. The function generator was set to a sweep time of 1999s. From these sweeps, resonances were found and recorded. Note that when searching for resonances between 7-10 Hz nonlinear and odd behavior was observed. After 3 times trying to get that data it was decided that these frequencies were too difficult to get and used 0.5-5 Hz for data analysis.

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