1.4 - Analyzing Displacement in Shallow Water
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Understanding Velocity Potential
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Today, we're discussing velocity potential, which is essential for calculating water particle displacement. Can anyone explain what velocity potential represents?
Isn't it a measure of the potential energy of fluid particles in motion?
That's partially correct! Velocity potential is actually a scalar function whose gradient represents the velocity of the fluid. Let's dive deeper into how this impacts displacement.
How do we express this mathematically?
Great question! We express velocity potential through integrals, which we will derive shortly.
To help remember, think of the acronym 'VAP' for Velocity, Area, and Potential. Keeping that in mind will help us with our calculations.
Displacement Equations in Shallow Water
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Now, let’s look at the equations for particle displacement in shallow water. Can someone tell me the relevance of the ratio of depth to wavelength?
It helps us understand whether we are in shallow or deep water conditions!
Exactly! If the depth-to-wavelength ratio is less than 1/20, we derive specific equations that show the particle displaces in an elliptical orbit.
And how do we write the elliptical equation?
We write it as Δx/D² + Δz/B² = 1. Keep this equation noted as it is foundational for understanding particle motion.
Deep Water Displacement
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Now let’s transition to deep water where the ratio d/L is greater than 1/2. How does that affect the particle's motion?
Does that mean the particles will move in circular orbits instead of elliptical?
Exactly! In deep water, both the D and B values play a crucial role and become equal, resulting in circular trajectories.
So the displacement equation becomes a circle equation, right?
Correct! This leads to an important equation you’ll need to remember: Δx² + Δz² = h² / e^(2kz).
Visual Representation of Displacement
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Visual aids can be really effective in understanding displacement. Can anyone describe the trajectory of water particles in different depth conditions?
In shallow water, they move in elliptical paths, and in deep water, they tend to move in circles.
Great summary! As the amplitude decreases with depth, the differences in trajectory become essential to study. Knowing how to visualize will help greatly in practical applications.
It sounds a bit like a wave-particle duality situation!
Indeed! It's vital to keep in mind the depth in studying displacement of water particles.
Introduction & Overview
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Quick Overview
Standard
The section explains how to calculate water particle displacement based on velocity potential and wave kinematic parameters, distinguishing the behavior of particles in shallow and deep water and deriving their trajectories.
Detailed
In this section, we analyze water particle displacement, focusing on the mathematical derivation of individual horizontal and vertical displacements using velocity potential. By establishing the relationships between these displacements and the parameters of wave motion, we demonstrate that water particles move in elliptical paths in shallow water and circular paths in deep water. The equations provided highlight the distinctions based on water depth, allowing us to conclude that the trajectories depend significantly on the ratio of the water depth to the wavelength.
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Understanding Water Particle Displacement
Chapter 1 of 5
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Chapter Content
We found 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.
Detailed Explanation
This chunk discusses how we calculate the displacement of water particles in a wave. Displacement refers to how far the particle moves from its original position. The first formula describes the displacement as an integral of velocity over time, integrating the product of the horizontal velocity (u) and a time function (g T w). The text also indicates that there are separate expressions for horizontal and vertical displacement, where the integral of u dt is important in calculating the exact movement of the water particle.
Examples & Analogies
Think of a water balloon on a flat surface. If you push down on one side, the water inside moves away from its original position. The amount it moves depends not just on the force you exert but also on how long you exert this force (time). In the ocean, virtual 'pushes' from waves make water particles move back and forth, and we can describe this movement mathematically.
Deriving Displacement Expressions
Chapter 2 of 5
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Chapter Content
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.
Detailed Explanation
Here, we see the derived expressions for horizontal and vertical displacements in terms of wave parameters (h, k, z, sigma, etc.). The horizontal displacement is given in terms of cosine functions, and the vertical displacement uses sine functions. The terms contain parameters that are determined based on wave properties, such as the wave height (h), wave number (k), and the vertical coordinate (z), indicating how these factors influence how far the particles move horizontally and vertically over time.
Examples & Analogies
Imagine you're on a swing at a playground. The swing moves back (horizontal displacement) and forth (vertical displacement) as you swing. The height of the swing represents the wave height (h), and the frequency of your swinging can be thought of as the wave's characteristics (k and sigma). Just like the swing's path, these mathematical expressions describe how water particles move under the influence of waves.
Shallow Water Conditions
Chapter 3 of 5
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Chapter Content
Now we analyze this displacement in shallow water. So what happens in shallow water for d by L less than 1 by 20... D becomes h by 2 into 1 by kd and B becomes h by 2 k d + z by kd.
Detailed Explanation
This part shifts focus to shallow water conditions, highlighting how specific ratios of depth to wavelength (d/L) affect the particle motion. In shallow water, certain approximations simplify the equations for D and B, which represent the semi-major and semi-minor axes of the elliptical motion of water particles. The result shows that under shallow conditions, the motion becomes more straightforward, with the displacements characterized by simpler ratios and factors associated with wave behavior.
Examples & Analogies
Consider a shallow pond versus a deep lake. In the pond, when you throw a pebble, the waves spread out quickly with fewer ripples, while in a lake, the ripples may travel farther without dissipating quickly. The analysis of water particles in shallow water is akin to observing how these ripples behave differently based on the depth of water.
Water Particle Movement in Deep Water
Chapter 4 of 5
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Chapter Content
In deep water for the case d by L greater than half D becomes h by 2 e to the power k d + z... this shows that for deep water condition the water particle trajectories are circular.
Detailed Explanation
This section discusses how particle motion differs in deep water compared to shallow water. The equations change based on the greater depth, leading to a circular motion for water particles rather than elliptical. The parameters adjust in such a way that the previously complicated displacements condense into a simpler form, indicating that deep water movement is different and more uniform than in shallow regions, depicted as circular paths.
Examples & Analogies
Picture a diving board at a swimming pool. When you dive into the deep end, your body moves through the water in circular arcs due to the wave patterns around you, akin to how water particles behave in deep water. On the other hand, in shallower parts, the water swirls differently, just as we saw in the earlier analysis of shallow water.
Implications of Water Particle Displacement
Chapter 5 of 5
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Chapter Content
The amplitude of the water particle displacement decreases exponentially along the water depth... This is important.
Detailed Explanation
This final chunk focuses on the implications of particle displacement concerning depth. As explained, the amplitude (extent of movement) decreases as you go deeper into the water. This concept is critical because it informs us about the wave energy distribution and how particles behave with respect to their environment, essential for understanding wave mechanics and the forces acting on them at different depths.
Examples & Analogies
Think of how sound travels in water. You can hear splashes near the surface clearly, but the sounds get quieter as you submerge deeper, similar to how wave amplitudes decrease. The higher energy waves above can't affect deeper water as much, just like the sound dissipates with distance.
Key Concepts
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Depth-to-Wavelength Ratio: The ratio affecting water particle motion.
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Displacement Equations: Representations of water particle movements in various depths.
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Elliptical and Circular Orbits: Distinct trajectories determined by depth.
Examples & Applications
Example of calculating horizontal and vertical displacements in shallow water based on given parameters.
Example of comparing the calculations for deep versus shallow water displacements.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
In shallow waters, particles sway, in ovals they dance, come what may.
Stories
Imagine a deep ocean where water particles dance in perfect circles, while in a shallow stream, they twirl in elegant ellipses.
Memory Tools
Remember 'SDE' for Shallow = Dance in Ellipses, Deep = Circular.
Acronyms
Use 'WAVE' - Water and Velocity Equations in displacement.
Flash Cards
Glossary
- Velocity Potential
A scalar function whose gradient represents the velocity field in fluid dynamics.
- Displacement
The movement of a particle from its original position, often described in horizontal and vertical components.
- Elliptical Orbit
The path that water particles follow in shallow water conditions, characterized by an elongated circular shape.
- Circular Orbit
The motion of water particles in deep water where they follow circular trajectories.
- DepthtoWavelength Ratio (d/L)
A dimensionless ratio that indicates the relationship between water depth and wavelength, affecting the nature of water motion.
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