2.2. Pressure Equation Derivation

So, you remember we talked about linearized Bernoulli’s equation, this equation is given by - del phi del t + p by rho + g z half rho squared + half u squared + w square we that term we avoided because of linearization.

The linearized Bernoulli’s equation is a simplification of the general Bernoulli equation, which describes the behavior of fluid flow. The equation is expressed in terms of the velocity potential, gravitational potential, and pressure. The term 'del phi del t' refers to the change in velocity potential over time, while 'p by rho' represents pressure per unit density. The terms involving squared velocity are neglected due to linearization, simplifying the calculation under specific conditions.

Imagine a river flowing steadily; if you want to predict how the water moves without considering every small ripple, you could use the simplified version of the Bernoulli equation much like how pilots use simplified flight paths for calculations, avoiding the complexities of every wind gust.

Now, if you multiply this throughout by rho the above equation, then the pressure can be given as you see, if we take beyond the other side it can be written as a rho del phi del t -.

Multiplying the linearized Bernoulli’s equation by the fluid density (rho) helps us isolate the pressure (p) within the equation. By rearranging the terms, we derive a new expression for pressure, which incorporates both dynamic and static components. The dynamic component (represented by 'rho del phi del t') captures the changes in fluid velocity, while the static component is associated with the pressure resulting from the water's depth (gamma z).

Consider a water balloon. When squeezed, the pressure inside increases (dynamic pressure). At the same time, the weight of the water at the bottom exerts a static pressure. By adjusting these elements in our equations, we can predict how the balloon will behave under different conditions, just like predicting water flow in rivers.

So, if we substitute for phi was the velocity potential in this equation, what do we get? p will be so del phi del t will be written as gamma h by 2 into cos h k d + z divided by cos h k d into sin k x - sigma t - gamma z so this is the pressure equation.

By substituting the velocity potential (phi) into the derived pressure equation, we express pressure explicitly in terms of wave characteristics like height (h), position (x), and time (t). The equation now encapsulates the dynamic behavior of the wave at various depths, reflecting how wave dynamics influence pressure at different points under the surface.

Think of how a sunset changes the colors in the water; light waves and reflections affect what we see below the surface. Similarly, this equation shows how changes in wave parameters can control pressure at different depths, revealing the hidden dynamics under the waves.

Now, it has to be mentioned that P was set to 0 to define the free surface boundary condition in Bernoulli’s equation if you remember, however, phi was determined by setting p = 0 as is that = 0 instead of z = eta.

Setting the pressure (P) to zero is crucial to defining the free surface boundary condition in fluid dynamics. This condition allows us to analyze how pressure varies in the fluid below the surface and serves as the reference point for measuring pressure changes at various depths. However, we define the velocity potential based on this condition, emphasizing that our derived equations only hold true below the water surface (z < 0).

Imagine the surface of a lake as a reference point where the water is calm (zero pressure). The deeper we go, the more the pressure due to the water above. This analogy helps translate the mathematical abstraction into tangible concepts that illustrate the relationship between depth and pressure in fluid mechanics.

So, the pressure distribution under progressive wave is given by this is crest wave crest, so, pressure distribution under wave crest and pressure distribution wave trough to this is the pressure distribution under a progressive wave.

The pressure distribution under a progressive wave varies between the wave crest and trough. At the crest, pressure is lower due to the uplift from below, while at the trough, pressure increases as the water is pushed down. Understanding this distribution is crucial for predicting structural responses in hydraulic engineering applications, such as naval architecture or coastal construction.

Picture standing on the beach watching the waves. At the top of the wave crest, your feet feel lighter, but when the wave settles, you feel the force of the water pressing down, illustrating how pressure changes dynamically as waves move.

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