1.1. Dynamic Boundary Conditions

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 learn about the governing equations relevant to dynamic boundary conditions. The primary equation mentioned is the Laplace equation, which is crucial in fluid dynamics. Additionally, Bernoulli's equation and the continuity equation are utilized because they describe how fluid flows and changes under different conditions. These equations help define specific boundary conditions necessary for understanding wave dynamics.

Think of the governing equations as the rules of a game. Just like in a game, players must follow certain rules to play effectively, fluids must adhere to the principles outlined in the governing equations to behave predictably.

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.

This chunk discusses how the dynamic boundary condition is applied to derive the value of phi 3. By implementing boundary conditions, we can express changes in fluid behavior—specifically, how wave potentials interact. The equation C = D * e^(2kd) suggests a mathematical relationship between the parameters of wave motion, with 'C' representing a dynamic quantity affected by variables such as water depth (d) and wave number (k).

Imagine setting up a trampoline. The height you jump is influenced by the tension in the trampoline fabric and the surface beneath you, similar to how 'C' is affected by 'D' and the wave conditions captured by 'k' and 'd'.

And same procedure is repeated for the dynamic free surface boundary condition and we get for phi 3 at term like this, you understand same procedure. So phi 3 we get - ag by sigma cos h k d + z similarly, we get phi 4.

Here, the procedure for applying dynamic boundary conditions continues, specifically regarding a free surface boundary condition. The formula phi 3 = -ag/σ cosh(kd) + z expresses the potential relating to factors such as wave amplitude (a), gravitational acceleration (g), and depth (d). This shows us how wave potential changes at the surface level, factoring in gravity and wave action.

Consider how a surfboard floats and moves on the surface of the water. The physics of waves (like those calculated with the formula) determine how high the surfboard sits and how it moves with the waves, just as dynamic free surface boundary conditions affect fluid behavior.

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 so this becomes, so if you add to velocity potential, this was you know, phi 2. And, this one with a negative sign was phi 1.

In this chunk, the discussion centers on how total velocity potential is calculated. It involves summing the various potentials derived earlier, specifically phi 2 and phi 1. The negative sign indicates that phi 1 represents a different directional influence possibly counteracting phi 2's effect. This calculation is essential in predicting how fluid behaves under dynamic conditions.

Think of it like a seesaw: if one side goes up (phi 2), the other side has to come down (phi 1) to maintain balance. The total movement (or velocity potential) is influenced by both sides working against each other, similar to how we calculate the overall flow dynamics.

On differentiating we can get this equation we differentiate we get k dx dt = we differentiate with respect to time = sigma or dx by dt can be written as sigma by k, and sigma was nothing but 2 pi by T and k was nothing but 2 pi by L.

This segment focuses on differentiating equations to determine the wave speed. By taking the derivative, we can relate wave numbers and frequencies through the equation dx/dt = σ/k. This relationship highlights how the wave speed (C) is calculated by dividing the angular frequency (σ) by the wave number (k), which connects wave motion with its physical dimensions.

Imagine timing a runner on a track. The runner's speed can be figured out by calculating how far they travel over time. Similarly, in wave mechanics, we can figure out wave speed by relating frequency and wavelength, much like measuring how quickly a runner completes a distance.

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

This chunk introduces the dispersion relationship, a fundamental concept in wave mechanics that describes how wave speed varies with wavelength and water depth. It forms a mathematical representation of how waves behave differently under various conditions, highlighting the relationship between wave characteristics and environmental factors such as water depth.

Think of a symphony orchestra where different instruments play at different pitches and volumes. In wave mechanics, just like how the varying strengths and tones in music create unique experiences, the dispersion relationship explains how different wave properties create distinct wave behaviors in water.

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