2.4. Pressure Response Factor (Kp)

So, to go ahead and get started with this topic of pressure distribution under progressive waves. (Refer Slide Time: 00:43) 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.

This introduction sets the stage for exploring how pressure is distributed in water waves. It links the topic to Bernoulli's equation, showing that pressure is influenced by water depth and wave characteristics. By addressing the linearization of the equation, the focus narrows to key terms, crucial for understanding waves.

Think about how waves break on the shore – depending on their height (like pressure) and how deep the water is (depth). The balance of forces at play can be compared to how you might feel different pressures while diving down in a pool—closer to the surface feels different than deeper water.

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 -. So, let me take this so, we have multiplied by rho so it becomes - rho del phi del t + p because rho gets canceled here + rho g z that and this can be written as gamma. So p can be written as if this goes on the other side, this becomes rho del phi del t and this one becomes - gamma z or + of - gamma z said this is the dynamic pressure.

This chunk describes the mathematical manipulations of Bernoulli's equation following multiplication by the density of the fluid, rho. It leads to the formulation of pressure, distinguishing dynamic pressure (due to wave velocity) from static pressure (related to water depth). Understanding these components is essential for analyzing wave behavior in hydraulic engineering.

Imagine squeezing a balloon. The tighter you squeeze (increasing dynamic pressure), the harder it becomes to push it down into water; that's similar to how dynamic pressure works in waves while maintaining a balance with static pressure from depth.

And this is the static component depending upon only the water depth now, 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.

This section connects the velocity potential, phi, to the pressure equation. By introducing terms involving depth and wave function parameters, it emphasizes the interdependence of pressure, fluid height, and wave dynamics. This relationship helps predict how pressure changes as waves interact with varying depths.

If you've ever watched waves roll in, the pressure at the bottom of the wave is more than at the surface because of the depth and height of the wave—similar to how a stronger wave's pressure makes you feel when underwater. This link aids engineers in designing structures like piers or hulls of boats.

So, Kp we say that Kp is pressure response factor, then we can simply write p as gamma eta into Kp - gamma z or p by gamma can be written as eta Kp - z.

The pressure response factor (Kp) represents how pressure changes with wave movement. It relates wave amplitude (eta) and depth (z), allowing for simpler expressions of pressure in waves. This factor is vital for accurate wave analysis and design in hydraulic systems, highlighting how pressure dynamics shift with wave characteristics.

Think of Kp as a feedback mechanism—just like how your voice might sound different in a cave compared to open air due to acoustics. Similarly, Kp measures how wave pressure responds dynamically based on the wave's height and depth at which you're measuring.

Now, if you apply this equation 2.57 so, pressure at z = 0 is going to be p by gamma = eta and pressure at z = - d so, this is free surface and this is at bottom z = - d at bottom...

This practical application explains how pressure varies at different water levels (at the surface versus at the bottom). Knowing these values is crucial for designing structures that can withstand the varying pressures encountered in both shallow and deeper waters.

When measuring tire pressure, you know the pressure at the surface (inside the tire) should be different from the pressure at deeper layers of the tire material. It’s similar here: understanding pressure variations in waves helps in designing sturdy ships or sea barriers.

So, this means that p by gamma is d - eta by cos h k d will always be greater than d - eta because this is greater than one; it is often needed to determine the surface wave height based on sub-surface measurement of the pressure.

This chunk discusses correction factors necessary for accurate surface wave height estimations based on measurements taken below the surface. It highlights the importance of Kp and the predictable variations in pressure, which are essential for safely designing marine structures and understanding wave behavior.

Consider how measuring water levels in a roller coaster ride versus at the bottom of the pool can yield different outcomes. This correction helps engineers design rides to ensure safety, similar to monitoring wave heights for coastal structures.

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

This section covers how pressure varies between the highest point (crest) and the lowest point (trough) of a wave. The distribution characterizes how waves exert different pressures at various heights, which is critical for understanding how to manage water structures.

Think of walking on sand—where the ground is compacted (the crest) is different from areas where the sand is less dense (the trough). Similarly, wave pressures can push harder or softer depending on their heights and depths.

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