2.1. Dynamic and Static Pressure Components

The pressure can be given as: p = rho del phi del t - gamma z. This is the dynamic pressure and this is the static component depending upon only the water depth.

In the context of wave mechanics, understanding the pressure in water involves two key components: dynamic pressure and static pressure. The equation p = rho del phi del t - gamma z defines these components. Here, 'p' represents the pressure, 'rho' is the density of the fluid, 'del phi del t' signifies a change in potential, and 'gamma' represents the weight density of water while 'z' is the depth. The term 'rho del phi del t' accounts for the dynamic component which varies with flow velocity, while 'gamma z' reflects the pressure from the weight of the water above a certain depth—this is the static pressure.

Imagine filling a glass with water. The pressure you feel at the bottom of the glass depends on how high the water is—the deeper you go, the more weight of water is pushing down. That weight creates static pressure. Now, if you start stirring the water, the pressure fluctuates because of the movement—this is similar to dynamic pressure. Think of it like the difference between standing still in a pool versus swimming; when swimming, you encounter changing pressures from the water movement around you.

Substituting for phi, p will be p = gamma eta K p - gamma z, where K p is the pressure response factor.

By substituting 'phi' with its relevant expression, which relates to the velocity of water, we derive another important form of the pressure equation. The term 'gamma eta K p' combines the wave amplitude (eta) with the pressure response factor (K p), allowing us to express pressure in terms of wave characteristics. Here, 'K p' encapsulates how pressure responds to changes in wave form and depth. Therefore, this form is crucial for understanding how pressure behaves in conjunction with waves as they propagate through the water.

Consider waves in the ocean as they approach the shore. When waves gather energy from the wind and build up, their height (or amplitude) increases—the pressure beneath that wave also increases. The 'K p' acts like a 'multiplier' that indicates how much more pressure is generated by that wave height at different depths. This is similar to feeling more pressure on your ears as you dive deeper into a pool; the deeper you dive, the greater the weight of water pushing down on you.

It has to be mentioned that P was set to 0 to define the free surface boundary condition in Bernoulli’s equation.

Setting 'P' to zero is critical in defining boundary conditions that characterize the behavior of fluids at their surface, particularly in wave mechanics. This is part of Bernoulli’s principle, where it’s important to understand that at the free surface of a body of water (where the water meets air), the pressure is atmospheric and can be treated as zero gauge pressure. Therefore, the equations that derive from this condition help inform predictions about how fluids behave in response to external forces, such as wind or grounding effects.

Think of the surface of a lake on a calm day. At the very top, the pressure you exert by placing a float is balanced by the atmospheric pressure—it feels like there is no weight because you're equalizing with the air above. This is akin to how we treat pressure at the boundary in our equations. Just like you would feel lighter while floating, the pressures balance out, allowing us to define behavior of waves correctly above the water surface.

Pressure at z = 0 is p/gamma = eta and pressure at z = -d relates to the behavior of pressures at wave troughs.

When we calculate pressure at depths, the relationship p/gamma = eta indicates that at the free surface (z = 0), pressure can be equated to the wave height. Conversely, observing pressure at z = -d allows us to determine conditions at greatest depths, specifically at the wave trough. This type of analysis is useful for understanding how wave movements influence underwater pressure and is critical in designing marine structures or understanding environmental impacts on ecosystems.

Visualize standing on the beach watching waves roll in. When a wave comes in, you feel its energy as it crashes at the shore (z = 0). If you were to dive underwater, you’d experience different pressures the deeper you go (at z = -d). Like in diving, at each depth you feel a different weight of the water above you—the pressure increases according to the height of the water above you.

--