1.2. Kinematic Bottom Boundary Condition

I do not expect you to remember the derivation but the steps you must know the things like the specifying the governing equation, what are the governing equations governing equation is Laplace equation for the boundary conditions we utilize the Bernoulli’s equation and the continuity equation for example, so, we have obtained phi 2 here.

In this chunk, we introduce the concept of governing equations relevant to kinematic boundary conditions. The governing equation in this context is the Laplace equation, which is crucial in fluid dynamics for analyzing flow situations. Alongside this, Bernoulli’s equation and the continuity equation are employed for setting boundary conditions. These equations help in solving problems involving fluid flow, allowing us to derive specific potential functions such as phi 2, which is used later in calculations.

Think of governing equations like the rules of a game. Just as players need to follow specific rules to ensure fair play, engineers and scientists must adhere to governing equations to accurately represent and predict fluid behavior. Just like understanding the game's rules helps you play better, knowing these governing equations helps in analyzing fluid dynamics effectively.

So, if we consider phi 3 and apply the same concepts that phi 3 = you know, we apply the dynamic boundary condition. This will give C equal to D into e to the power 2 k d.

Here, we delve into dynamic boundary conditions. When calculating phi 3 (a potential function), we use previously established concepts. The dynamic boundary condition allows us to analyze how the fluid responds dynamically to changes in conditions, leading to a formula: C = D * e^(2kd). This formula gives insight into how certain parameters interact within the fluid system, showcasing the dynamic behavior that is influenced by depth (d value) and wave number (k value).

Imagine a rubber band stretching when you pull it. The way it reacts to the force you apply reflects a dynamic condition - similar to how fluid behaves under certain conditions. By applying a dynamic boundary condition to our calculations, we can predict how 'stretched' or altered the fluid's potential becomes, akin to measuring the extension of the rubber band.

Now you remember I said that the total velocity potential will be the summation of the two terms. So our velocity potential is going to be phi 2 - phi 1, or also in terms of phi 3 and phi 4 also.

In this segment, we explore how total velocity potential is calculated through the sum of different phi terms. The relationship shown here indicates that our total velocity potential is derived by subtracting phi 1 from phi 2, though it can also incorporate phi 3 and phi 4. This calculation is essential for modeling how fluid flows and responding to wave conditions, utilizing the results from earlier derivations based on boundary conditions.

Think of creating a multi-layered cake, where each layer represents a different phi value (i.e., phi 1, phi 2). When you combine the layers (subtracting one from another), you create the total delicious cake — the velocity potential in fluid dynamics. The act of combining these individual phi values helps you understand the complete picture of fluid behavior, just like how each layer contributes to the final taste of the cake.

So, to determine the velocity of the wave, this is how the wave celerity that is length wave length by time period is the celerity and this is the basis of finding the way of celerity that if locate a point and traverse along the wave such that at all time t our position relative to the waveform remains fixed when this will happen when we are going to move with the speed of the wave.

This chunk highlights the concept of wave celerity, defined as the wave length divided by the time period. Celerity indicates how fast a wave travels through a medium. To maintain the same position relative to the wave, one must move at the same speed as the wave itself. This relationship is essential in applications ranging from engineering to oceanography, where understanding wave behavior can influence design and predictions.

Consider riding a wave while surfing. To stay on the wave without falling, you need to match its speed. If you go too slow, you'll fall behind; if you speed up, you might wipe out. This real-world experience of adjusting your speed captures the essence of wave celerity — understanding how fast the wave is moving is crucial to successfully riding it.

Now, one important thing after this derivation of the velocity potential is something called a dispersion relationship that is one of the core concepts of wave mechanics.

The dispersion relationship links wavelength, wave period, and water depth, forming a pivotal concept in wave mechanics. This relationship is developed under the assumption of small amplitude waves, which states that the wave slopes are minimal. This principle allows us to derive the famous equation that defines the behavior and characteristics of waves as they propagate through different water conditions.

Imagine tossing a stone into a calm pond. The ripples that spread out are akin to waves, and each ripple affects the others, changing their speed and direction slightly as they travel through the water. The dispersion relationship captures this effect mathematically, showing how depth influences wave behavior — similar to how deep water may allow for larger, slower-moving waves compared to shallower areas.

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