1.1. Kinematic Parameters

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. 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.

This chunk discusses how we begin with the concept of velocity potential, which describes the velocity of particles in waves. Velocity potentials are mathematical functions representing how a wave travels through a fluid. The mention of changing parameters highlights that different calculations can be done using various functions. We ultimately aim to derive the water particle displacement, which is determined through integration of the velocity function over a time interval, providing insights into how far and in what direction particles move in a wave.

Imagine a surface of a lake during a windy day. The wind creates ripples on the surface that have a specific pattern of movement. The velocity potential refers to how we would calculate the speed and direction of those ripples, much like predicting the path of a moving object based on its speed and direction.

So, the expression of individual horizontal and vertical particle displacement is integral u dt u we already know in terms of h before. So, the final results comes out to be h by 2 cos hkd + z phi by sin hkd into cos k x - sigma t. Similarly, the vertical displacement delta z is given by h by 2 sin hkd + z into sin hkd into sin k x - sigma t these are the periodic terms. So, once you derive the velocity potential everything can be found out.

This chunk presents the specific equations for calculating the horizontal and vertical displacements of water particles in waves. The horizontal displacement is expressed in terms of various parameters such as wave height (h), wave number (k), and time (t). The vertical displacement is similarly derived and includes sine functions, which indicate the periodic nature of waves. These equations reflect how particles move in relation to the wave, demonstrating that their movements can be predicted mathematically once we have the velocity potential.

Think of a child on a swing, which goes back and forth (horizontal) and up and down (vertical). By knowing how high the swing rises and how far it goes forward, we can predict its path using equations similar to how water particle movements in waves are predictable through the sinusoidal functions in the equations.

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. So, what we write is delta x by D whole squared + delta z by B whole squared = 1, this is in general what type of equation if D is not equal to be elliptical. So, this is an equation of analysts showing that water particle moves in in an elliptical orbit.

This section explains how the motion of water particles in waves can be modeled using the equation of an ellipse. Here, delta x and delta z represent the horizontal and vertical displacements, respectively, while D and B refer to the respective maximum displacements. The sum of the squares of the displacements divided by their maximum values equals one, a characteristic property of ellipses. This illustrates that, under certain conditions, water particles follow elliptical paths that are influenced by their depth and the wave conditions.

Consider a figure skater spinning on the ice. As they extend their arms, they trace out an elliptical path with their movements. Similarly, water particles in waves describe elliptical orbits due to the forces acting upon them, such as wave motion and gravity.

Now we analyze this displacement in shallow water. So what happens in shallow water for d by L less than 1 by 20 we have 0.05 we have used cos hkd + z and sin hkd + 2 sin hkd + z goes to k d + z and sin hkd goes to kd.. Hence, D will be what if we as you remember D was this equation D was h by 2 cos hkd + z and sin hkd in shallow water this becomes kd + z divided by kd.

This chunk focuses on how the behavior of water particles changes in shallow water conditions. It states that when the ratio of depth (d) to wavelength (L) is less than a specific threshold (1/20), certain simplifications apply to the equations of motion. For shallow water, the previously derived equations are modified, and D becomes proportionate to the ratio of depth to wave number, providing a clearer understanding of particle behavior in these conditions.

Imagine walking in a swimming pool. As the water is shallow, your movements cause more noticeable waves that travel differently than if you were in the deep end. Just like how our analysis changes based on water depth, so does the behavior of underwater particles in waves when conditions change from deep to shallow.

But in deep water for the case d by L greater than half D becomes h by 2 e to the power k d + z + e to the power - kd + z divided by e to the power kd. - e to the power - kd as D is very large e to the power - k d + z and e to the power - kd will be very small correct because this deep now, so, if it - sin they will be very small compared to e to the power k d + z. So, this can be taken away.

In this chunk, we analyze particle displacement in deep water. When the ratio of depth (d) to wavelength (L) exceeds a certain proportion, the behavior of the displacement equations simplifies significantly, emphasizing the exponential factors associated with wave heights and depths. The small exponential terms can be ignored, allowing us to focus on how water particles move in circular orbits due to the symmetry in the equations, simplifying the analysis of their movement.

Think of a large amusement park wave pool where the waves are substantial. In the deep end, the waves create circular patterns, similar to how a person floating on a large inner tube would move in a circle without much resistance from the underlying water. The deeper areas lead to circular orbits for the water particles, resulting in a more consistent movement pattern.

So, 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.

This section encapsulates how we visually represent fluid particle trajectories. As we go deeper into the water, the amplitude of water particle displacement diminishes due to various factors, including gravitational forces and wave dynamics. The trajectories, thus become less pronounced with depth, showcasing how water particles near the surface experience greater displacement due to wave action compared to those deeper down.

Imagine tapping the surface of a calm pond; the ripples expand outward but fade as they move away from the initial point. The deeper you go under the surface, the less you feel those ripples. In the same way, water particle movement diminishes with depth beneath the waves, demonstrating a decrease in amplitude of their movement.

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