1.5. Final Velocity Potential Formula

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. 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.

In this section, we discuss the governing equations for wave motion, specifically focusing on the Laplace equation, Bernoulli’s equation, and the continuity equation. These equations are crucial for determining the potential functions (phi 2, phi 3) that describe the flow dynamics in a fluid. The dynamic boundary condition is applied to calculate phi 3, and various relationships arise from this. Understanding these conditions is crucial for grasping how fluid moves under different forces.

Think of water flowing over a dam. The Laplace equation would help model how the water behaves at various points, just like understanding how the pressure changes at different depths. When you throw a stone in a pond, the ripples that form are influenced by the relationships that these equations describe, demonstrating the effects of boundary conditions.

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, cos k x cos sigma t + sin k x sin sigma t can be return as cos k x - sigma t using trigonometry. So, the final velocity potential with the value I meant the formula for which you are supposed to remember is this 1 ag by sigma cos h k into d + z divided by cos h k d into cos k x - sigma t this is the final velocity potential.

The velocity potential is established by summing contributions from different potentials (phi 2 and phi 1). The final formula incorporates the results of trigonometric manipulations, which reveals a relationship between various parameters like amplitude (ag), angular frequency (sigma), and depth (d). This provides a complete expression that describes the fluid flow behavior under wave motion.

Imagine creating waves in a pool as you jump in. The collective disturbances (or potential functions) represent how energy translates through the water, similar to how our equation captures these interactions. The final formula serves as a way to calculate the effects of those interactions on the wave patterns you see.

So, to determine the velocity of the wave, this is the so, 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.

Wave celerity (C) is defined as the speed at which a wave travels across a distance over a specific time period. The relationship between wave length (L) and the time period (T) helps to calculate celerity. If you follow a wave while moving at its same speed, your position remains constant relative to it. This illustrates how celerity is crucial in understanding wave dynamics in various media.

Consider surfing: to catch a wave, you must paddle with the wave’s speed. If you’re too fast or too slow, you'll miss it. This idea directly relates to celerity; you need to be at just the right speed (C = L/T) to remain in sync with the moving wave.

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. So, the relationship between wavelength with period and water depth is obtained as given below for the dispersion relationship, the main assumption while establishing the relationship is that since we are dealing with small amplitude waves.

The dispersion relationship describes how waves of different wavelengths travel at different speeds, depending on the depth of the water. This relationship is central in wave mechanics, indicating that longer waves travel faster than shorter waves in shallow water, a phenomenon observed in nature. Understanding this helps to predict how waves behave under various conditions, especially as they interact with their environment.

Think of a line of children jumping into a pool at differing intervals. The waves they create will travel along the surface at different speeds depending on how far apart they jump. In a similar way, the dispersion relationship helps us understand why waves change speed based on their length and depth.

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