4. Wave Energy

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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