1.6. Particle Trajectories in Deep Water

So, in this case eta will be a cos k x - sigma t, the u will change u and a x and w and a w, so, if we assume a different velocity potential you remember we had to velocity potentials. So, we started doing all the calculation with the first velocity potential but instead of the first day we started with the second wewill obtain this set of the wave kinematic parameters.

This chunk introduces the concept of wave kinematic parameters, referencing the velocity potentials used in the calculations of water particles in motion. The velocity potential is a function that helps describe the speed of water particles based on a sinusoidal wave model represented as eta. Understanding how these parameters change based on different scenarios (like using a second potential) is fundamental for analyzing water particle behavior in waves.

Think of the velocity potential as a recipe for a smoothie. Depending on whether you use strawberries (first potential) or mangoes (second potential), the final taste (particle behavior) of your smoothie will change. Similarly, different choices in velocity potential will alter the calculation outcomes for water particle movements in waves.

So, the expression of individual horizontal and vertical particle displacement is integral u dt u we already know in terms of h before. So, the final results comes out to be h by 2 cos hkd + z phi by sin hkd into cos k x - sigma t. Similarly, the vertical displacement delta z is given by h by 2 sin hkd + z into sin hkd into sin k x - sigma t these are the periodic terms.

In this section, we break down the mathematical expressions that describe how water particles move in both horizontal and vertical directions during wave action. The displacement is expressed in terms of integrals, and the final expressions show that particle displacement is periodic, a key characteristic of wave motion. The horizontal and vertical components are connected to wave height and depth through trigonometric functions, revealing the cyclical nature of these displacements.

Imagine you are riding a bicycle over hills. As you move up and down, your vertical height changes (like vertical displacement), and your position on the ground changes horizontally (like horizontal displacement). The mathematical relationships here represent how a water particle behaves similarly under wave forces, cycling through both dimensions.

So, we have proved particle moved 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 chunk discusses the nature of the particle trajectories in water. It establishes that, during wave motion in deeper waters, particles generally move in elliptical orbits due to the differences in horizontal and vertical displacements. The semi-major axis (D) indicates how far particles move horizontally, while the semi-minor axis (B) denotes the vertical movement. Understanding this motion is necessary to grasp how waves interact with particles in the water.

Consider a child swinging on a swing set. When they swing forwards and backwards, their path is not a perfect circle, but an elongated oval (ellipse). The horizontal distance they swing out is further than how high they swing up and down. Similarly, water particles’ movement can usually be described using ellipses when waves pass through.

Now we analyze this displacement in shallow water. So what happens in shallow water for d by L less than 1 by 20... Hence, D will be what if we as you remember D was this equation D was h by 2 cos hkd + z and sin hkd in shallow water this becomes kd + z divided by kd.

The text discusses how particle movements change from deep to shallow water. In shallow water, where the depth (d) is much less than the wavelength (L), the expressions for D and B simplify significantly. This results in different orbital shapes and characteristics that define how particles act under different environmental conditions. The discontinuity between shallow and deep waters affects water particle trajectories, indicating different fluid dynamics scenarios.

Think about swimming in a shallow pool versus in the ocean. In a pool, you can reach the bottom easily, causing less disturbance to the water above. In contrast, when swimming in the ocean, the waves push you around more due to the deeper water's properties. Likewise, particle movements are influenced by the water's depth.

Therefore, the particles move in circular orbits in deep water since these equal to be with the equation of the form delta x divided by h by 2 e to the power kz + delta z h by 2 e to the power kz whole squared = 1.

In deep water, the analysis reveals that the particle motion simplifies to circular orbits. This happens because in deep water conditions, the distinctions between horizontal and vertical displacements equalize, forming a circular motion. The equation provided reflects this relationship mathematically, indicating that both the horizontal (delta x) and vertical (delta z) displacements equalize under specific conditions.

Visualize a buoy floating on the surface of deep water; as waves pass by, it moves in a circular manner while keeping its position intact. This buoy’s behavior represents how water particles in deep water take on circular orbits, demonstrating the impact of wave motion on stationary objects.

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 equalto one half the wavelength below this still water level.

This chunk explains how the impact of a wave diminishes with depth in the ocean. As you go deeper, the amplitude of water particle movement decreases exponentially, which means that the effect of the surface waves is felt less strongly below a certain depth (half the wavelength). This principle is crucial for understanding wave energy transmission and water dynamics.

Imagine standing in a pool during a wave-making event. The closer you are to the surface, the more the waves jostle you. However, if you dive deeper, the action of the waves feels less and less intense. This reflects how wave energy diminishes with depth, and below a certain point, it feels almost calm.

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 speaker sets the stage for the next part of the course, indicating that future discussions will focus on concluding the analysis of wave mechanics, particularly transitioning to pressure distribution and progressive waves. This shows how understanding water particle trajectories paves the way for analyzing other phenomena in fluid dynamics.

Like finishing a chapter in a book, the speaker is ready to transition to new concepts. Just as closing one set of ideas prepares us to understand others more deeply, grasping wave mechanics enhances our understanding of other fluid concepts in engineering.

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