7.1. Linearization of the Bernoulli's Equation

Now, if you linearize the Bernoulli’s equation, so linearizing means taking out the second order term. So, we take out u squared + w squared because those are quite small at the free surface. So, this is the linearized boundary condition that we linear linearized Bernoulli’s equation that we get.

Linearizing an equation means simplifying it by removing higher-order terms that have minimal effect on the results. In this case, we are focusing on Bernoulli's equation, which relates the speed, pressure, and height of fluid flow. At the free surface, the velocities (u and w) are relatively small, so the terms involving u squared and w squared can be considered negligible or removed to simplify the analysis.

Imagine you are cooking and need to soften a tough piece of meat by simmering it for a long time. At the start, if you check the meat too frequently, the boiling water might splash around and create a mess. Instead, you could let it simmer without constant checking. In fluid mechanics, we streamline the process by ignoring small effects (like u squared and w squared) to focus on the main factors affecting the flow, just like letting your meat simmer without disruptions.

So when the height will be z = eta that means, that free surface if we take phi = 0 equation in equation 2.4 we are going to get eta = 1 by g del phi del t at z = eta. So, in terms of velocity potential we have obtained by linearizing the Bernoulli’s equation we have obtained you see why we have assumed u squared + w squared = 0.

In fluid dynamics, when we linearize Bernoulli's equation, we establish a relationship that describes how the height of the fluid surface (denoted as 'eta') evolves over time. By assuming that the velocity potentials are small, we can express 'eta' as a function of time derivative of the potential divided by gravitational acceleration (g). This relationship simplifies our understanding under the assumption that velocities are much smaller than the fluid motion driven by gravity.

Think of throwing a small pebble into a calm pond. The initial splash is minimal, so we can assume that the waves created by the pebble's impact are small compared to the overall stillness of the water. Similarly, in our calculations, we assume the speed is small, allowing us to use simple relations to describe how the water level changes when disturbances occur.

So, this is the eta = So, eta which was unknown from before using the dynamic 3 surface boundary condition this dynamic 3 surface boundary condition, we have been able to write eta as a function of velocity potential.

The assumption of small amplitudes in wave motion means that the height (eta) of the waves is quite small compared to the wavelengths involved. By applying the dynamic surface boundary condition, we can derive a clear mathematical expression for 'eta' based on the velocity potential. This relationship allows us to predict how the free surface will vary and reinforces our understanding of fluid dynamics under these conditions.

Consider ripples in a pond caused by a gentle wind. The height of each ripple is much smaller than the distance between them. This observation allows us to simplify equations that describe the water's surface since we can treat the ripples as negligible in height compared to the overall surface of the pond. In the same way, we ignore higher wave heights when calculating their effects on fluid motion.

Therefore with this assumption of that the waves being as small. This data can be written as, instead of 0 = eta we write z = 0 because eta is very small, so it will be z = 0.

By considering the amplitude of the waves to be very small, we simplify our analysis even further by stating that the wave height can effectively be treated as close to zero (z = 0). This assumption helps in solving complex equations associated with fluid movement, ensuring that the primary factors influencing the fluid's behavior are easily manageable.

Think about a tiny hill in a flat field—a hill might be present, but when looking over vast fields, it appears almost flat and negligible. Similarly, in fluid dynamics, if the waves are tiny compared to the scale of interest, we can treat them as if they don't significantly alter the flatness of the surface we are analyzing, allowing us to focus on broader patterns in fluid movement.

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