Wave Energy
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Introduction to Wave Energy
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Welcome class! Today we will be focusing on wave energy. Can anyone tell me what wave energy refers to?
Is it the energy carried by waves?
Exactly! Wave energy is essentially the energy transported by waves. It's composed of both potential and kinetic energy. Any guess on how we can express that mathematically?
Isn't it something like potential energy plus kinetic energy?
Right! So, the total wave energy can be calculated using the formula: Total Energy = Potential Energy + Kinetic Energy. This is foundational for understanding wave dynamics.
Potential and Kinetic Energy in Waves
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Let's delve deeper. The potential energy of a wave is based on the height of the water column above a reference depth. Can anyone define how we calculate this?
We can integrate the potential energy over the height of the wave above a reference depth.
Correct! We use the formula for potential energy as it varies with the wave height and depth. For instance, it can be expressed as Gamma * A^2 / 4, where A is the amplitude of the wave.
And what about kinetic energy?
Great question! The kinetic energy is expressed as Kinetic Energy = 1/2 * mass * velocity^2. In the context of waves, we can derive it using the velocities in wave motion. Remember, both energies contribute equally to the total energy.
Pressure Distribution in Waves
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Now, let's transition to pressure distribution under progressive waves. Who can summarize how pressure relates to wave height?
Pressure seems to vary based on water depth and wave height.
Exactly! The pressure is calculated using the formula p = gamma * (eta * Kp - z). Where Kp is the pressure response factor, eta is the wave height, and z is the depth. What does this imply for diving below the free surface?
As we go deeper, the static pressure increases, and we can derive the exact pressures beneath wave crests and troughs.
Group Velocity and Wave Energy Significance
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Finally, let’s go into group velocity. Why do you think understanding group velocity is important in the context of wave energy?
Because it affects how waves travel and accumulate energy in certain areas?
Correct! Group velocity describes how wave groups propagate and is different from individual wave speeds. The formula CG = C/2 for deep water shows that group velocity is half the wave speed. It's crucial for predicting energy distribution as waves reach coastlines.
So, it helps in understanding potential energy accumulation in shallow waters!
Exactly! Always remember how group velocity helps us gauge where energy will concentrate.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
Wave energy encompasses both kinetic and potential energy found in progressive waves. By subtracting the potential energy in the absence of waves, we can derive the energy solely due to wave action. Key equations for calculating pressure distribution, energy density, and the impact of wave phenomena such as group velocity are explored.
Detailed
Detailed Summary
In this section, we delve into wave energy, primarily focusing on how to compute total energy represented as the sum of potential and kinetic energy in water waves. We start by examining the pressure distribution formulas derived from wave mechanics, such as Bernoulli's equation and its linearized forms. The section progresses into detailing how the potential energy above a certain depth, combined with the kinetic energy from wave motion, significantly contributes to understanding wave energy.
The computation of energy is intricate and divided into two components: potential energy due to water displacement caused by waves, and kinetic energy resulting from the movement of water particles in the wave. The total energy of waves can be simplified to express both components and then analyzed to provide a practical formula governing energy due to waves. Additionally, concepts like group velocity and pressure response factors (Kp) are discussed, providing vital information necessary for coastal and hydraulic engineering applications.
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Understanding Wave Energy Components
Chapter 1 of 8
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Chapter Content
Total energy potential energy + kinetic energy. So, in order to determine the total energy and a progressive wave the potential energy of the wave above z = -d with a wave from present is determined from which the potential energy of the water in absence of the wave is subtracted.
Detailed Explanation
Wave energy consists of two primary components: potential energy and kinetic energy. To find the total energy associated with a progressive wave, we first calculate the potential energy of the wave at the surface above a reference point (denoted as z = -d). This includes the energy contributed by the wave itself. We then subtract the potential energy of the same volume of water in the absence of the wave. The difference gives us the energy that is solely due to the presence of the wave.
Examples & Analogies
Imagine a water bottle sitting at the edge of a table. The potential energy in this case is the energy stored due to its height above the ground. If someone shakes the bottle (simulating a wave), the energy of the shaken liquid is different from when it is still. By measuring how much energy is added by the shaking, we can understand the wave energy represented here.
Calculating Potential Energy
Chapter 2 of 8
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The potential energy with respect to z = -d of a small column of water d + Eta high dx long and 1 meter wide can be written as, so if we have a small column of water d + eta high, where dx is length and 1 meter wide potential energy will be gamma a x bar area will be d + eta into d + eta by 2 into dx.
Detailed Explanation
To calculate the potential energy of a small column of water influenced by wave height 'eta', we consider a column with height (d + eta), where 'dx' represents its horizontal length and is 1 meter wide. The integral for potential energy involves the density of water (gamma) multiplied by the height of the column and its position. The potential energy is therefore expressed mathematically based on these values.
Examples & Analogies
Think of this calculation as measuring how much water you have in a tall glass. The higher the water is above a certain level (like how we measure wave height), the more potential energy the water has. If you increase the height either by adding more water (wave) or tilting the glass, you are increasing the potential energy.
Averaging Potential Energy
Chapter 3 of 8
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The average potential energy per unit surface area is going to be an integration of gamma by 2 into so, we have to average over the whole wave period and also average over the entire wavelength.
Detailed Explanation
Calculating the average potential energy per unit surface area requires integrating the potential energy over a complete wave cycle or period, as well as across the entire wavelength. This means we take into account how the potential energy varies as the wave oscillates up and down and average it out, ensuring that we capture the energy contribution from every part of the wave.
Examples & Analogies
Imagine you are measuring the average height of waves over an entire beach. Instead of just taking a snapshot at one moment, you would stand at the shore and watch how high the water rises and falls over time. By doing this, you get a more accurate average height of the waves, which relates to their average potential energy.
Energy with and without Waves
Chapter 4 of 8
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However, this is the average potential energy per unit surface area of all the water above z - z = -d. Now, we should also calculate the potential energy in absence of the wave and the wave is not there, when the wave is not there then there will be no eta in this term you see there was eta here.
Detailed Explanation
When calculating the potential energy, it's essential to differentiate between the scenario with waves present and one where waves are absent. When waves are absent, the potential energy calculation excludes the wave height 'eta', resulting in a lower value. The difference between these two energy states helps us isolate the contribution of the wave itself.
Examples & Analogies
Returning to our water bottle analogy, think of the water in the bottle with a lid (the wave presence) versus a completely still bottle without any motion (absence of waves). When the water is still, it has a certain level of potential energy. Once you start shaking the bottle (introducing waves), the potential energy increases because of the additional movement.
Final Calculation of Wave Energy
Chapter 5 of 8
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The total energy due to the presence of the wave will be the total energy - the energy that are - the energy when the wave is not there. This gives us the potential energy due to waves is gamma a square by 4 where a is the wave amplitude.
Detailed Explanation
Finally, to determine the energy solely attributed to the waves, we subtract the potential energy without waves from the total energy with waves. This calculation leads us to understand that the potential energy due to waves can be expressed as gamma a^2 / 4, where 'a' is the amplitude of the wave, providing a clear equation that expresses this energy relation.
Examples & Analogies
Consider a pendulum swinging. The total energy of the pendulum at the top of its swing (maximum height) is at its peak due to gravity. When it swings down to its lowest point, most of the energy is kinetic. The energy difference at the peak versus when it is hanging still represents the potential energy added by the swinging motion, similar to how we calculate wave energy.
Kinetic Energy in Waves
Chapter 6 of 8
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Now we will also need to find kinetic energy it is very simple kinetic energy is half mv squared, where m is the mass of the fluid and v is a result and velocity.
Detailed Explanation
Kinetic energy in the context of waves can be expressed using the basic formula of kinetic energy, KE = 1/2 mv², where 'm' is the mass of the fluid involved in the wave motion and 'v' is the velocity of the waves. This kinetic energy can be gathered from moving water, where the flow velocity contributes to the energy.
Examples & Analogies
If you observe a flowing river, the energy of the water moving quickly represents kinetic energy. If the river slows down (like in calmer conditions), the kinetic energy reduces. The energy in flowing water is akin to that in waves, which also have mass and speed, contributing to their kinetic energy.
Relationship Between Potential and Kinetic Energy
Chapter 7 of 8
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Chapter Content
So we can show that kinetic energy is gamma a squared by 4 potential energy was also gamma a squared by 4. Therefore, 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
Finally, it is established that both kinetic and potential energy for waves can be calculated to be equal in magnitude, amounting to gamma a² / 4 for each. When combined, they sum up to total energy due to the waves, which equals gamma a² / 2. This demonstrates the energy dynamics within wave motion, showcasing how these components interact.
Examples & Analogies
Think of a swing in a playground; at the highest point, it has maximum potential energy. As it swings down and picks up speed, it has kinetic energy. At the bottom of the swing, it has its highest kinetic energy and lowest potential energy. This cycle is very similar to how energy interchanges between potential and kinetic components in waves.
Energy Density of Waves
Chapter 8 of 8
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Chapter Content
So, the average total energy per unit surface area is the sum of average potential and kinetic energy density is often called as a specific energy density.
Detailed Explanation
The average total energy per unit surface area of a wave represents the energy density, which combines both kinetic and potential energies considered over a wave cycle. This specific energy density is crucial for understanding the energy output that can be harnessed from waves and how it varies with wave characteristics.
Examples & Analogies
Consider how a solar panel captures energy from sunlight. Just as the panel collects energy over its surface area, the water's energy can be viewed as energy density collecting over the ocean's surface as waves pass by. Knowing the energy density helps us in designing efficient systems to capture and use this energy.
Key Concepts
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Wave Energy: The energy carried by waves, resulting from both kinetic and potential components.
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Kinetic Energy: Energy of fluid particles in motion, significant for wave dynamics.
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Potential Energy: Energy stored due to a fluid column's height, influencing wave energy calculations.
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Group Velocity: The speed at which a group of waves travels through water, essential for energy transport understanding.
Examples & Applications
Example of calculating wave energy using specific wave height and amplitude.
Illustration of how pressure distribution affects the buoyancy of floating objects on waves.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Wave energy, oh so grand, potential and kinetic, hand in hand.
Stories
Imagine a surfer riding waves. The higher the wave, the more energy supports the rider, reflecting the potential energy, while the moving water exemplifies kinetic energy.
Memory Tools
PEKE (Potential Energy + Kinetic Energy = Wave Energy) helps remember that both forms contribute to total wave energy.
Acronyms
WAVE stands for Wave Energy
'What Am I Value Every?' to remember energy encompasses both motion and height.
Flash Cards
Glossary
- Wave Energy
The energy carried by waves, composed of kinetic and potential energy.
- Potential Energy
The stored energy in a water column due to its height relative to a reference point.
- Kinetic Energy
The energy of motion associated with water particles in a wave.
- Pressure Response Factor (Kp)
A factor indicating how pressure responds in relation to wave height and depth.
- Group Velocity
The speed at which the overall shape of a wave group travels, distinct from individual wave speeds.
Reference links
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