2.3. Free Surface Boundary Condition

Now, it has to be mentioned that P was set to 0 to define the free surface boundary condition in Bernoulli’s equation. However, phi was determined by setting p = 0 as is that = 0 instead of z = eta. Hence this means this particular equation is valid only for negative z. This means we cannot apply this equation here in this domain because it was derived with certain approximations and assumptions.

In fluid mechanics, the free surface boundary condition describes the behavior of fluids at the surface, where pressure is often set to zero against the atmosphere. In this context, it states that when setting the pressure (P) to zero at the free surface, we assume the velocity potential (φ) is adequately defined. This implies that the calculations for pressure are only valid for points below the free surface (negative z values). It emphasizes that the equations derived apply under certain conditions and assumptions. Therefore, care must be taken not to apply these derived equations above the free surface where they do not hold.

Imagine you are standing in a swimming pool. Above the water (the free surface), the pressure of air is different from the pressure experienced by your feet underwater. At the surface, the pressure you feel is atmospheric (essentially zero in relation to the water), whereas as you dive deeper, the pressure on your body increases. The equations that define pressure in the pool water are only applicable beneath the surface where the water's weight and depth create pressure. Above the water surface, those same equations won’t apply.

So, if you apply this equation on the pressure at z = 0 (the free surface) is going to be p by gamma = eta. Pressure at z = -d (the bottom) under wave conditions means, if we n by cosh(kd) + d we know that it is definitely less than d + eta because cosh(kd) is always greater than one.

In this scenario, we can express the pressure at the free surface using the variable η, which represents the surface elevation due to the waves. At the free surface (where z = 0), the pressure divided by the specific weight (γ) of the fluid equates to the surface elevation η. When considering points deeper in the fluid (at z = -d), we use a modification of the equation to reflect the increasing pressure due to the weight of the water column above. This relative pressure distribution helps predict how pressure changes with wave activity and depth.

Consider how a diver feels pressure when swimming. Near the surface, the water is not heavy on them because not much water is above them. As they swim deeper, the pressure of the water increases significantly due to the weight of the water above them. The calculations discussed help understand this dynamic and relate it to surface wave conditions, enabling engineers to predict how waves will impact structures like boats, piers, and coastal defenses.

It is often needed to determine the surface wave height based on subsurface measurement of the pressure. So, if we have subsurface measurement techniques available, we can calculate n, where n is the correction factor depending upon the period, depth, wave amplitude. n should be greater than one for long-period waves and less than one for short-period waves.

In practice, engineers can measure pressure at various depths to infer information about wave heights and characteristics at the surface. This is crucial for ensuring that coastal structures, such as breakwaters and piers, can withstand the energy exerted by waves. The correction factor (n) accounts for how pressure readings can vary depending on whether the wave period is long or short, which in turn affects the accuracy of the wave height estimations derived from those readings.

Imagine scientists using underwater microphones (hydrophones) to track ocean waves. By measuring the pressure fluctuations underwater, they can estimate how high those waves are at the surface. Just as you might need a different tool to measure a small stream versus a large river, these scientists adjust their calculations based on how long or short the waves are, ensuring they provide accurate information about water conditions that could affect boats or coastal construction.

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