1.5. Deep Water Conditions

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 we will obtain this set of the wave kinematic parameters.

In this chunk, the discussion begins with the variable 'eta', which is related to wave motion and is expressed as a function of position (x) and time (t). The variables 'u', 'a', 'w' represent different physical parameters associated with wave behavior. The speaker notes the importance of velocity potentials, showing how different assumptions (starting from one velocity potential over another) affect the calculations of wave parameters, which are essential for understanding wave dynamics.

Think of velocity potential like a recipe for making a cake. There are many ways to make a cake (using different ingredients or methods), and each way results in a slightly different cake. By choosing a different approach or ingredient (velocity potential) in wave calculations, you can understand various outcomes in wave behavior.

So we have studied the velocity potential, we found out the velocities under the progressive where we have found out the acceleration now importantly we have to find out water particle displacement is nothing but integral u times g T w times d T in extend that direction respectively. So, the expression of individual horizontal and vertical particle displacement is integral u dt u we already know in terms of h before.

The chunk discusses how to calculate water particle displacement based on water velocity. By integrating the velocity over time ('u dt'), one can determine how far these particles move. The terms 'horizontal' and 'vertical' displacement relate to how water particles shift within the wave, which are further described by specific mathematical expressions involving wave height (h) and other parameters.

Imagine you're floating in the ocean on a surfboard. As waves come, your board moves both up and down (vertical displacement) and back and forth (horizontal displacement). Understanding how far and fast you move with the wave helps you predict your position on the water’s surface.

Therefore, we can write cos squared k x - sigma t is delta x by d whole squared, let us go back. So, what we have said is this is D and this is B. So, if we take D down so, it will become delta x by d = cos k x - sigma t and if we take D down here it becomes delta z by D = sin k x - sigma t. So, sin squared theta + cos squared theta = 1 therefore, this is what we have used.

In this section, the speaker describes how to express the horizontal (delta x) and vertical (delta z) displacements in terms of cosine and sine functions, indicating that the combination of these displacements reveals a circular or elliptical movement. The relationships used here rely on fundamental trigonometric identities, which show that particle movement can be represented mathematically in various forms.

Visualize a child on a merry-go-round. As they rotate around the center, they move both forwards and upwards depending on their position. This relates to how water particles move in circular or elliptical paths when waves pass through them, with the mathematical relationships representing their precise movements.

Therefore, you see, this is the representation the schematic representation of fluid particle trajectories 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 equal to one half the wavelength below this still water level that is the deep water.

The discussion highlights how, as you dive deeper into the water, the influence of wave motion decreases. Water particle displacements shrink significantly compared to the height of the waves nearer to the surface. This behavior is visualized through a schematic that illustrates particle trajectories, emphasizing the diminishing impact of waves as depth increases.

Consider a boat bobbing on the surface of the ocean during a storm. As you dive deeper, you’ll notice that the water is calmer and the waves feel less intense. This illustrates how wave impact weakens with depth, similar to how particles beneath a certain depth in water feel less motion compared to surface waves.

Therefore, particles move in circular orbits in deep water since these are 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.

This concluding aspect reveals that in deep water, the fluid particles follow circular paths rather than elliptical ones. This is mathematically represented by an equation which explains the relationship between horizontal and vertical displacements and confirms the nature of motion in deep water conditions.

Think of a washing machine on the spin cycle. The clothes (water particles) are forced to move in a circular path due to the machine’s rotation. In deep water, like the washing machine's spin, movement is more uniform and circular as opposed to irregular paths that may occur in shallower waters.

--