4.2. Potential Energy Calculation

Total energy = potential energy + kinetic energy. To determine the total energy in a progressive wave, the potential energy above z = -d is calculated, subtracting the potential energy of water in the absence of the wave.

In fluid dynamics, particularly in wave mechanics, we define total energy as the sum of potential energy and kinetic energy. When waves occur, the potential energy is influenced by the height of the water column above a certain reference point (z = -d). To find the energy of the wave itself, we subtract the potential energy of the water at rest (i.e., without the influence of waves) from this total. This separation allows us to isolate the potential energy that is directly due to the presence of the waves.

Think of a water tank. If it is filled with water at a certain level, it carries potential energy due to gravity. When waves move through the tank, they raise the water level even higher at certain times. The energy associated with this increased height, compared to when the water is still, is what we mean by the potential energy of the wave.

Potential energy for a small column of water can be written as: $PE = \gamma (d + \eta)^2 / 2 \times dx$. Average potential energy per unit surface area is obtained by integrating over the whole wave period and wavelength.

To calculate the potential energy of a small column of water with a height adjusted to include wave effects, we use the formula $PE = \gamma (d + \eta)^2 / 2$, where $d$ is the depth of water at rest and $\eta$ is the wave height. After setting this up, we must average this energy over one full cycle of the wave (wave period) and over the spatial length of the wave (wavelength) to get a representative value for all waves.

Imagine measuring the energy stored in a series of waves at a beach. If we consider just one wave, we can see how high it lifts the water. To find the average energy for all waves that hit the beach in a set time (like a tide cycle), we would need to account for each wave’s height and combine them mathematically, just like calculating the average height across several hills.

When the wave is not there, the potential energy simplifies to: $PE_{no wave} = \gamma d^2 / 2$. The energy due to waves is then: $PE_{waves} = PE_{with waves} - PE_{no waves} = \gamma a^2 / 4$.

Next, we need to define the potential energy when there are no waves present. This energy is simply calculated based on the still water level. The potential energy due to the waves can then be found by subtracting the energy without waves from the energy with waves. This leads us to a final formula for the potential energy due to waves alone, showing that it is proportional to the square of the amplitude of the waves.

Think of a trampoline. When nobody is on it, it’s at rest and has a certain potential energy based on its height. When someone jumps, they increase the height and the trampoline's potential energy changes. If we want to calculate how much extra energy is added when someone is jumping (the waves), we subtract the original height's energy from it to determine the extra energy contributed by the jump.

The average kinetic energy per unit surface area can also be calculated and shows that kinetic energy equals potential energy: $KE = \gamma a^2 / 4$. Therefore, the total energy becomes: $TE = KE + PE = \gamma a^2 / 2$.

The kinetic energy in a wave, which is the energy due to the movement of water, can also be derived and is found to be the same as the potential energy calculation. Both kinetic and potential energies can be computed to arrive at the total energy for waves. The result indicates that energy in waves is distributed equally between potential and kinetic forms, leading to a complete view of the energy dynamics in wave mechanics.

Imagine a swing at a playground. When the swing is at its highest point, it has a lot of potential energy, which converts to kinetic energy as it swings downwards and speeds up. The total energy of the swing remains constant because if one type of energy goes up, the other goes down, just like in waves where potential energy and kinetic energy interchange.

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