Introduction to wave mechanics (Contd.)
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Pressure Distribution under Progressive Waves
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Today, we will discuss the pressure distribution under progressive waves, an essential concept in wave mechanics. Can anyone describe what happens to pressure as waves travel through water?
I think the pressure changes based on the depth and the height of the wave?
Exactly! The pressure at any point can be calculated using the modified Bernoulli equation, which accounts for dynamic and static pressure based on wave height and water depth.
What do you mean by dynamic and static components?
Good question! Dynamic pressure is related to wave motion, while static pressure is influenced solely by water depth. Let's write down an equation: p = ρ(∂φ/∂t) - γz.
Can we use this equation for all depths?
Only below the free surface, as implied from our assumptions during linearization.
Why is it important to separate these pressures?
Separating them helps us understand how waves interact with the water and enables engineers to design structures accordingly.
In summary, pressure under waves varies with both wave characteristics and water depth, crucial for calculating the impact on structures in hydraulic engineering.
Understanding Group Celerity
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Now, let's explore the concept of group celerity. How do you think the speed of a group of waves differs from individual waves?
Are you saying that when waves travel together, they don't just move at the same speed as alone?
That's right! When wave groups travel, their speed depends on factors like wavelength and wave amplitude. Can you think of a real-world analogy for this?
Like running with friends? You can't run as fast as when you're alone?
Exactly! The same principle applies to wave groups. Understanding group velocity is essential for predicting wave impacts on structures.
What are the formulas to calculate this velocity?
For shallow water, we often use CG = C, while for deep waters, CG = C0/2. Remember, C refers to wave speed, and these formulas help when designing coastal structures.
So to wrap up, the group velocity varies with wave interactions, which is crucial for maintaining the integrity of coastal and hydraulic structures.
Calculating Wave Energy
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Next, let’s dive into wave energy. What do you think makes up the total energy of a wave?
It must be the sum of potential energy and kinetic energy?
Precisely! The total energy of progressive waves can be derived from both energy forms. How can we represent them in an equation?
If potential energy is γa²/4, then kinetic energy is also γa²/4?
Spot on! Therefore, the total energy E can be expressed as γa²/2, which is important for engineers to know for structure safety.
How do we get these values during real applications?
We typically measure wave heights, periods, and depths to calculate these energy values accurately while ensuring that ground structures are safeguarded against potential wave impacts.
In summary, understanding wave energy contributes significantly to efficient hydraulic engineering designs and ensures resilience in coastal environments.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section builds upon previous discussions of wave mechanics by detailing the relationship between pressure and wave dynamics, including pressure distribution in progressive waves, the significance of group celerity, and the calculations for wave energy. The reader is introduced to important equations and interpretations necessary for hydraulic engineering applications.
Detailed
Detailed Summary
This section of hydraulic engineering focuses on progressive wave mechanics, specifically on pressure distribution and energy calculations. The introduction elaborates on the linearized Bernoulli’s equation, where the pressure can be expressed in terms of dynamic and static components, influenced by water depth and wave potential.
Key Points:
- The section reaffirms the Bernoulli’s equation and how it has been adapted through linearization to express pressure dynamics influenced by wave interactions.
- The equations for pressure response include variables like the velocity potential (φ), wave amplitude (η), and depth (z), leading to subsequent calculations of total energy related to the wave.
- The concept of group celerity is introduced, explaining how wave trains, due to superposition, will have a different velocity than individual waves. This emphasizes its importance in practical applications, especially in calculating long and short-period wave characteristics.
- The discussion stands on the fundamental understanding of energy calculations where potential and kinetic energy are contrasted to derive equations governing wave energy, crucial for applications in coastal and hydraulic engineering.
- Important results are highlighted, showing average potential energy and kinetic energy in progressive waves, underscoring the energy relationship in various water depths.
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Pressure Distribution Under Progressive Waves
Chapter 1 of 6
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Chapter Content
So, to go ahead and get started with this topic of pressure distribution under progressive waves. So, you remember we talked about linearized Bernoulli’s equation, this equation is given by - del phi del t + p by rho + g z half rho squared + half u squared + w square we that term we avoided because of linearization. Now, if you multiply this throughout by rho the above equation, then the pressure can be given as you see, if we take beyond the other side it can be written as a rho del phi del t -.
Detailed Explanation
The discussion starts with the foundation of pressure distribution in progressive waves using a modified form of Bernoulli’s equation. This equation describes fluid motion and characteristics under wave action. By multiplying the linearized Bernoulli equation, we incorporate density (rho) to express pressure in terms of potential (phi) and dynamic (static pressure components) relations.
Examples & Analogies
Think of a water wave as a moving car pushing water ahead of it. Just like how the pressure changes in front of the car compared to the back, the pressure at different points under a wave changes based on the wave's dynamics and the water's depth.
Dynamic and Static Pressure Components
Chapter 2 of 6
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So becomes p = rho del phi del t - gamma z or + of - gamma z said this is the dynamic pressure. And this is the static component depending upon only the water depth... However, phi was determined by setting p = 0 as is that = 0 instead of z = eta. Hence this means this particular equation is valid only for negative z.
Detailed Explanation
In the modified equation, p represents the total pressure, which is a combination of the dynamic pressure (from the wave motion represented by del phi del t) and the static pressure component (dependent solely on the water depth, represented by -gamma z). The condition p=0 is set to define the boundary at the water surface.
Examples & Analogies
Imagine a swimmer diving underwater. The deeper they go, the greater the pressure they feel from the water above them. The surface pressure can be likened to the static component, while the pressure caused by the wave above represents dynamic pressure.
Pressure Response Factor Definitions
Chapter 3 of 6
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So p by gamma can be returned as eta K p - z... Now, it has to be mentioned that P was set to 0 to define the free surface boundary condition in Bernoulli’s equation.
Detailed Explanation
This statement introduces the concept of pressure response factor (Kp). Kp takes into account the effects of the wave on the pressure felt at different depths. By expressing pressure as a function of wave height (eta) and applying the boundary conditions, we can assess how pressure responds in relation to wave dynamics. The significance of setting pressure to zero helps establish conditions at the free surface of the water.
Examples & Analogies
Consider throwing a rock into a still pond. The ripples created are like waves, and the pressure exerted at various depths changes based on how high the water is pushed up by the ripples. This resembles the pressure changes described with Kp and can help us understand how waves exert force underwater.
Group Celerity Concept
Chapter 4 of 6
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Now, there is something called group celerity, a very important concept. So, when a group of waves or a wave train travels its speed is generally not identical with the speed of the individual waves.
Detailed Explanation
Group celerity is the speed at which a group of waves moves, distinct from the speed of individual waves within that group. This difference in speed is crucial for understanding the overall behavior of waves in fluids, as the collective motion of waves can affect energy transfer and can lead to phenomena such as wave interference.
Examples & Analogies
Think of a parade of floats moving down a street. Each float can move at its own pace (like individual waves), but as a whole, the parade (the group of waves) might move at a different speed. This concept illustrates how group dynamics can lead to differing velocities in water waves.
Wave Amplitude and Nodes
Chapter 5 of 6
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Thus, the amplitude of the wave and this is the phase and the amplitude varies from 0 to 2 or depending upon the, what the phase is here. So the point of 0 amplitude are called nodes. They can be located by finding the 0s of the cosine factor.
Detailed Explanation
In wave mechanics, the amplitude of a wave, which indicates its height, can change based on the wave's phase. Nodes are points where the amplitude reaches zero, which occur due to the superposition of different waveforms. Understanding where these nodes exist is vital for grasping how wave patterns form.
Examples & Analogies
Consider a jump rope being swung in a wave motion. Points where the rope does not move up or down at all ('zero amplitude') correspond to nodes. This visualization helps in understanding how such points manifest in water waves.
Wave Energy Calculation
Chapter 6 of 6
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Now, we must go towards wave energy. So, total energy potential energy + kinetic energy... So, the total energy is going to be due to the waves alone is potential energy + kinetic energy, the total energy is gamma a squared by 2.
Detailed Explanation
Wave energy is the sum of its potential energy and kinetic energy, tracking how energy is transmitted through water. The derived equations illustrate that energy is proportional to the square of the wave amplitude, emphasizing that even small increases in wave height can result in substantially more energy. Therefore, in the analysis of waves, understanding energy dynamics is essential.
Examples & Analogies
Imagine a water wheel driven by wave action. The energy transformed to the wheel allows it to turn and do work, similar to how wave energy drives physical processes in marine environments. The total energy can influence activities like electricity generation from ocean waves.
Key Concepts
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Pressure Distribution: Variation of pressure in the water column influenced by waves.
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Bernoulli’s Equation: Fundamental equation in fluid dynamics relating pressure, velocity, and elevation.
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Dynamic Pressure: Pressure related to the motion of fluid.
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Static Pressure: Pressure attributed only to depth in the water column.
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Group Velocity: Speed of wave energy propagation.
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Potential Energy: Energy due to position in a gravitational field.
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Kinetic Energy: Energy associated with motion.
Examples & Applications
Calculating pressure at a depth of 5 meters under a wave of height 2 meters.
Estimating group velocity for a train of waves in a coastal engineering scenario.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
In wave mechanics we find, pressure's depth-designed, dynamic moves with speed, static's stable indeed.
Memory Tools
To remember energy forms in waves: 'PK for Total Energy' — Potential and Kinetic together.
Stories
A wave crest rode high, while beneath it, the water held pressure low. The wave traveled with friends, not too fast, not too slow, learning of pressures high and low.
Acronyms
PDE for Pressure Dynamics in Energy
Potential
Dynamic
Energy!
Flash Cards
Glossary
- Bernoulli’s Equation
An equation that describes the conservation of energy in fluid dynamics and relates pressure, velocity, and elevation.
- Pressure Distribution
The variation of pressure within the water under the influence of waves and depth.
- Dynamic Pressure
The component of pressure corresponding to the kinetic energy of fluid motion.
- Static Pressure
The pressure associated with the fluid's potential energy, dependent only on its depth.
- Celerity
The speed of wave propagation through the water.
- Group Velocity
The speed at which the energy or information is propagated by a group of waves.
- Potential Energy
The energy held by an object due to its position relative to a gravitational field.
- Kinetic Energy
The energy that an object possesses due to its motion.
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