2.2. Differentiation of Wave Parameters

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 chunk, we start with the derivation of velocity potential by considering phi 2 and moving on to phi 3. The dynamic boundary conditions are applied, which leads to the equation phi 3 = C = D * e^(2kd). This reflects a methodical approach in wave equations where we derive potential for waves based on their boundary conditions.

Think of this like adjusting the water level in a bathtub. The water level acts similarly to our wave potential, and how much water affects the level is like the boundary conditions we apply here to derive our equations.

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

We re-confirm that the method of applying boundary conditions is consistent. By applying the dynamic free surface boundary conditions, we derive phi 3 and subsequently phi 4 using similar steps. The negative signs in expressions can represent properties like inertia in moving fluids.

Imagine you’re stacking blocks. Each layer of blocks represents computing each potential. The last block placed on top (phi 4) is affected by the layers below it, similar to how boundary conditions influence each phi in our equations.

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.

This chunk covers how the total velocity potential is derived from combining different phi terms. We emphasize the relationship between phi 2, phi 1, phi 3, and phi 4 and how they contribute to the overall formula for velocity potential.

Consider a recipe for a cake where you need to mix different ingredients (the phi terms). Each ingredient adds flavor or texture, similar to how each phi term contributes to the total velocity of our wave scenario.

And the speed with which we must move to accomplish this will be given by k x - sigma t = constant or k x = sigma t + constant.

This part explains how to relate wave speed to the concept of keeping a constant phase. When you traverse along a wave, maintaining your position relative to it leads us to the conclusion that the wave speed, termed celerity, can be calculated as the ratio of wavelength to its period.

Imagine riding a bike next to a track where waves are moving. To maintain your position on the wave crest, you adjust your speed to match that of the wave. This is a practical example of how celerity operates – if you slow down or speed up without matching the wave's speed, you’re no longer at the crest.

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 describes how the wave's speed (celerity) changes with frequency and wavelength in a fluid medium. It is essential for understanding how different factors like water depth affect wave behavior. This relationship is formally derived using the derived equations and provides a link between the variables involved.

Think of a symphony orchestra where different instruments (representing different wavelengths) play at distinct speeds and volumes. Just like how some instruments harmonize better at certain pitches, waves have their own way of interacting, leading to the concept of dispersion.

--