1.7. Dispersion Relationship

So, we have studied the velocity potential, we found out the velocities under the progressive where we have found out the acceleration now importantly we have to find out water particle displacement is nothing but integral u times g T w times d T in extend that direction respectively. So, the expression of individual horizontal and vertical particle displacement is integral u dt u we already know in terms of h before.

In this part, the text discusses the importance of understanding how water particles move under wave action, specifically their displacement. The displacement is calculated using integrals of the velocities found earlier. Essentially, we are learning how far and in what direction the water particles move due to wave energy.

Imagine throwing a stone into a pond. The ripples you see are like waves, and as they travel outward, the water particles move up and down and side to side. Just as the ripples create patterns, the formulas we’re discussing help us understand how the water responds to those waves.

So, what we write is delta x by D whole squared + delta z by B whole squared = 1, this is in general what type of equation if D is not equal to be elliptical. So, this is an equation of analysts showing that water particle moves in an elliptical orbit. Here D is the semi major axis at the horizontal measure of the particle displacement and B is these semi minor axis that is the vertical measure of the particle displacement.

This equation relates the horizontal (delta x) and vertical (delta z) displacements of water particles, forming an elliptical shape. Here, D represents the 'semi-major axis' or the horizontal extent, and B represents the 'semi-minor axis' or the vertical extent of the orbit. This indicates that as waves propagate, water particles do not just move up and down but also sideways, tracing an elliptical path.

Think of an ellipse as the orbit of a planet around the sun but on a smaller scale with water particles. Just like a planet moves both closer and farther from the sun, the water particles oscillate in an elliptical motion in response to wave energy rather than simply up and down.

Now we analyze this displacement in shallow water. So what happens in shallow water for d by L less than 1 by 20 we have 0.05 we have used cos hkd + z and sin hkd + 2 sin hkd + z goes to k d + z and sin hkd goes to kd..

In shallow water conditions, the behavior of water particle displacement changes. The text presents formulas that modify D and B based on the shallow water depth. As the wave height and wavelength change, the relationship between these measurements affects the motion of the water particles, transitioning from elliptical orbits to simpler forms.

Imagine walking through a shallow pool versus a deep lake. In shallow water, you can feel your feet on the bottom, which limits how much you can move. In waves, this simplifies the motion of water particles, similar to how you would walk more easily on shallow ground compared to deep water.

So, in deep water for the case d by L greater than half D becomes h by 2 e to the power k d + z divided by e to the power k d - e to the power - kd as D is very large e to the power - k d + z and e to the power - kd will be very small.

In deep water, as the depth of water increases, the motion of water particles simplifies further as they begin to move in circular orbits rather than elliptical. The equations demonstrate how, in deep water, the changes in wave height and particle movement conform to predictable patterns, which mean water particles move in a circular path.

Consider how a roller coaster spins around a track. In deep water, the water particles are akin to the coaster cars moving in a circular loop, showing that even in deep water, the wave action maintains a consistent circular pattern.

So, you see, this is the representation the schematic representation of fluid particle trajectories the amplitude of the water particle displacement it first of all it decreases exponentially along the water depth the water particle displacement becomes small relative to the wave height at a depth equal to one half the wavelength below this still water level.

This section highlights how particle displacement diminishes with depth, becoming less pronounced compared to the surface wave height. This understanding is crucial in applications, as it shows how wave energy dissipates as you go deeper in the water. The visual representation aids in grasping the concept of how movement diminishes below the surface.

Imagine tossing a stone in the ocean. The ripples you see at the surface are large and energetic, while deep below, the movement is less noticeable, just like how the distance from the waves affects how much the water moves.

So, I think this is a fine point to stop. In the next lecture, we will conclude this module of invested flow that is wave mechanics and we start with the pressure distribution and the progressive waves from the next lecture and finish this module.

The concluding remarks indicate that the dispersion relationship has been discussed alongside its implications for wave mechanics, setting up the expectation for the next topic, which will delve into pressure distributions. This transition highlights the interconnectedness of these concepts in fluid dynamics.

Just like chapters in a book build on each other to create a coherent story, the learning about wave mechanics prepares us for understanding how pressure systems develop in fluid applications. Each idea works like a stepping stone.

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