1.1. Lecture # 63

Welcome back student to the last lecture of this module and also the last lecture of this course of hydraulic engineering. So, to go ahead and get started with this topic of pressure distribution under progressive waves.

In this introduction, the professor welcomes students to the final lecture, setting the stage for discussing pressure distribution in water waves. Understanding how pressure changes underneath waves is crucial for studying fluid dynamics in hydraulic engineering.

Think of waves in the ocean like a blanket on a bed. Just as the blanket applies pressure evenly across the bed, waves apply pressure on the water beneath them, affecting various factors in hydraulic systems.

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.

Here, the professor references the linearized Bernoulli's equation, which simplifies the understanding of fluid flow under wave conditions. The equation factors in various forces acting on the fluid, helping to predict how pressure behaves as waves move through water.

Consider how a straw works when you sip a drink. The pressure differences created by your sucking affect the liquid flow, similar to how the pressures in the equation affect water motion during wave occurrence.

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 -.

This step involves manipulating the equation to derive terms for pressure in a wave context. By multiplying through by the fluid density, the equation rearranges to isolate pressure, which is critical for understanding dynamic and static forces on water.

Imagine trying to rearrange furniture in a tight space. Similarly, manipulating the equation helps us better understand how components of pressure interact within the confines of fluid dynamics.

So p can be written as rho del phi del t - gamma z or + of - gamma z said this is the dynamic pressure. And this is the static component depending upon only the water depth.

In this section, the professor explains dynamic and static pressure components. Dynamic pressure relates to wave movement, while static pressure depends on the water depth. Recognizing these differences is crucial for engineers designing structures that interact with water.

Think of how a building's foundation is affected by wind (dynamic pressure) versus the weight of the building itself pressing down into the soil (static pressure). Similarly, water wave pressures impact structures differently based on their type.

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

This section discusses how the free surface boundary condition is crucial for correctly applying Bernoulli's equation in wave mechanics. Setting pressure to zero on the surface simplifies calculations and assumptions.

Consider a swimming pool: the water's surface is usually calm and defined. By stating pressure is zero at this boundary, we can make accurate predictions of how waves behave above this 'calm' surface.

So, we can write and K p we say that K p is pressure response factor...

The pressure response factor (K_p) relates how pressure changes in response to wave movement with respect to depth. Understanding this term is key for engineers to gauge pressures exerted by waves at various depths.

Think of this like a sponge reacting differently when squeezed harder or softer. Just as but depends on pressure exerted, water seeks stability while influenced by varying wave conditions.

Now, if you apply this equation 2.57 so, pressure at z = 0 is going to be p by gamma = eta...

This section explains how to calculate pressure at different depths using the derived equations. By applying differing values of z, engineers can predict how wave pressures change from the surface downwards, which is essential for structural integrity.

Think of the varying pressures felt at different depths if you dive into a swimming pool. Just like how you feel the pressure on your ears increase, similar calculations help predict pressures felt by structures underwater.

So, the pressure distribution under progressive wave is given by this is crest wave crest...

In this section, the professor outlines how pressure differs when analyzing wave crests and troughs. The mathematical model demonstrates nuanced changes in pressures, informing predictions for engineers.

Consider the differences in force you feel if you lay on a trampoline when it’s pushed down (trough) versus when it rebounds (crest). This analogy highlights how pressure varies based on wave positions.

Now, there is something called group celerity, a very important concept...

The professor introduces the concept of group celerity, which refers to the speed of wave groups traveling together as opposed to individual wave speeds. Understanding this difference is crucial for studying wave behaviors in different conditions.

Visualize a crowd of people running at different paces: the crowd as a whole (group) moves at a different speed compared to individuals running faster or slower. Similarly, waves traveling together exhibit unique dynamics.

So, if we find the x node by dt is the wave group velocity CG...

This section dives deeper into calculating the speed of wave groups. By deriving equations related to wave nodes, students gain insight into how groups of waves interact and the implications for coastal engineering.

Think about how waves roll into shore together, creating larger swells. The combined motion of these waves illustrates how wave groups translate differently than single waves do over time.

Now what is wave energy? We have discussed about almost everything...

The professor transitions into discussing wave energy, highlighting the total energy contributions from potential and kinetic components in waves. This understanding is vital in assessing energy generation from tidal and wave systems.

Just like how waves can be harnessed to generate electricity in tidal systems, understanding wave energy provides insights into conserving and utilizing natural forces efficiently.

So how do we determine total energy and the way we determine the total energy and we total energy...

Here, the professor explains how to derive both potential and kinetic energy from waves using mathematical formulas. This duality highlights the importance of both energy types in understanding wave dynamics.

Consider how a swing moves: at the highest point, it has maximum potential energy. As it swings down, that potential energy transforms into kinetic energy. Just like swings, waves change between these energy forms.

Therefore, the total energy is going to be due to the waves alone is potential energy + kinetic energy...

This part summarizes that total energy, derived from the earlier calculations, helps understand energy conservation in waves. Such insights assist in optimizing designs for buildings in coastal regions.

The way we store energy in a battery also reflects how energy in waves can be effectively harnessed. Balancing potential and kinetic forms creates opportunities for utilizing waves for energy needs.

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