4.4. Total Energy from Potential and Kinetic Energy

In order to determine the total energy in a progressive wave, the potential energy of the wave above z = -d with a wave present is determined from which the potential energy of the water in absence of the wave is subtracted.

Total energy in a wave context refers to the combined potential and kinetic energy that a wave carries. To calculate this, we first identify the potential energy present when a wave is generated, specifically considering the height above a reference level (z = -d). We then look at how this differs when the wave is absent, providing us with the energy attributed solely to the wave.

Imagine a beach ball floating on water. When the ball is submerged, some potential energy is stored based on its height above the base level of the water. If we remove the ball, we would measure the energy of the water without the ball, thus isolating the ball's contribution to energy in its buoyant state.

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 express the potential energy of a specific column of water influenced by the wave, we consider the volume of this column (1m width and varying height due to the wave, represented as 'd + eta'). The average potential energy can calculated using the height, multiplied by gravitational force (gamma) multiplied by the effective height and cross-sectional area.

Think of stacking blocks on one side of a table. The higher you stack, the more energy is contained in that stack due to its potential energy. Here, we calculate energy based on how tall the water column is because a taller column exerts more force — just like stacking blocks higher requires more effort.

The average potential energy per unit surface area is going to be integration of gamma by 2 into so we have to average over the whole wave period and also average over the entire wavelength.

Calculating average potential energy requires integration across one complete wave cycle (wave period) and across the entire length of the wave (wavelength). This averages out the potential variations over time and space, giving a more accurate depiction of average potential energy experienced across the surface.

It's like measuring the temperature in a room. Instead of checking just one corner, you take multiple readings throughout the space. By averaging those readings, you get a true sense of the temperature in the entire room, rather than just one hotspot.

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.

Kinetic energy in wave dynamics is determined by the fluid motion involved in the wave. By subtracting the potential energy of water when at rest (absence of the wave), we can quantify the kinetic energy produced by the motion of water molecules as the wave propagates through. This gives a clear view of how much energy is mobilized due to the wave's movement.

Consider a flowing river. When the water is still, it has potential energy due to its height. As the water flows and creates ripples, it gains kinetic energy from the movement. The waves are like the flowing action of the water — by measuring it as they flow, we can isolate their unique contribution to total energy.

The total energy is going to be due to the waves alone is potential energy + kinetic energy. The average total energy per unit surface area is the sum of average potential and kinetic energy density.

The total energy carried by waves is the sum of both kinetic and potential energy contributions. By understanding how these two forms of energy interplay, we can determine how much energy is actually transferred through a wave, which is critical for applications in engineering, dam design, and coastal management.

Think of total energy like the energy of someone running. It consists of potential energy (if they’re standing at the top of a hill) and kinetic energy (while they’re running down). The faster and higher they run, the more total energy they have in motion. Just as waves carry energy, so does a runner down a hill!

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