was plotted against depth 
and compared with the theoretical shallow water approximation curve as seen in Figure 1.
Phase Speed and Depth
The relation between phase speed and depth was explored in shallow water waves. A small wave tank (30 cm across) was filled with water at depths of 1.1, 2.0, 3.0, and 6.0 cm. At each depth, the tank was raised about a centimeter and set down to create a wave. The time it took the wave to travel across the tank ten times was recorded and divided from the total distance travelled of 300 cm to find the phase speed of the wave. Wave phase speed was plotted against depth and compared with the theoretical shallow water approximation curve as seen in Figure 1.
The dispersion relation shows how temporal and spatial frequencies of the surface waves are related,
where 
is the wavenumber. When wavelength 
is long compared to the depth,
and we get the shallow wave approximation, where the wave speed is proportional to depth
Figure 1. Phase speed vs depth and the theoretical shallow water approximation
In a large tank (1 m long), we attempted to produce approximate deep water waves by creating waves by hand with a wooden block. Four different frequencies were approximated, measuring the wavelength of each. The depth of water was 15 cm, so to successfully generate deep water waves, we would have had to produce 0.015 m wavelengths. This would require frequencies on the order of kilohertz, which is not possible to produce by hand. A longer and significantly deeper tank, with a wave machine that could consistently and quickly produce waves may allow for a closer approximation of true deep water waves.
Figure 2 shows the theoretical dispersion relation of waves, where shallow water waves would be observed when the depth of the water is approximately 7% of the wavelength and deep water waves would occur as described above. Given the size of the tank and water depth, neither true shallow or deep water waves could be produced.
Figure 2. Theoretical dispersion relation and experimental values based on wavelength measurements. True deep water waves would fall around a wavenumber of 400, and given the plotted limit of shallow water waves, it is clear the waves we produced were in between true shallow and true deep water waves.
Wave Particle Motion
Particle motion in waves was demonstrated by adding dye vertically throughout the water column and observing its behaviour as waves passed through it. For deep water waves, particle movement dissipates with depth and there is no motion at the bottom of the water column. The shapes formed by the dye at the surface reveal circular particle orbitals. In shallow water, movement was evident throughout the water column wherein the waves moved the dye into more elliptical shapes. While neither type of wave was truly achieved, there was enough distinction to observe this difference in particle paths.
Wave interaction with the tank along with probable irregular hand movement to create waves resulted in nonlinear waves and the Stokes drift effect. At the end of the passage of a sinusoidal wave, fluid particles return to their original position – orbitals are closed. For nonlinear or finite amplitude waves (Figure 3), near-surface fluid particle paths are no longer closed orbits (Figure 4) and there is a net transport of particles in the direction of wave propagation known as Stokes drift. This phenomenon can be seen in the deep-water dye demonstration in Figure 5.
Figure 3. Nonlinear wave behaviour through time [Fluid Mechanics (Kundhu & Cohen), p378].
Figure 4. Particle orbit for nonlinear waves [Fluid Mechanics (Kundhu & Cohen), p383].
Figure 5. Deepwater particle movement. On the right, circular dye shapes have formed near the surface showing dye movement at the top and not the bottom. Stokes drift can also be observed as the dye is moving to the left, the direction of wave propagation.
Wave Packets
We attempted to create wave packets by creating 3 waves close together, then waiting approximately 1 second before creating another 3 waves. We were briefly able to observe that the group speed was higher than the phase speed, as the wave crests quickly formed and died out. While phase velocity was easier to observe in this case, it is only the speed at which the wave packet appears to move. The rate at which the waves are actually moving is the group velocity, which determines the propagation rate of energy.
It was difficult to observe an isolated wave packet before the first crest reached the end of the water tank. We presume this would be easier to observe in a longer water tank. Figure 6 depicts wave packets travelling along x similar to what was observed, though the packets are more distinct in the graph.
Figure 6. Propagation of wave packets [https://www.fiberoptics4sale.com/blogs/wave-optics/100149702-phase-velocity-and-group-velocity]
