1.2. Introduction to wave mechanics (Contd.)

So, to go ahead and get started with this topic of pressure distribution under progressive waves. 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. 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 -.

The discussion starts with the foundation of pressure distribution in progressive waves using a modified form of Bernoulli’s equation. This equation describes fluid motion and characteristics under wave action. By multiplying the linearized Bernoulli equation, we incorporate density (rho) to express pressure in terms of potential (phi) and dynamic (static pressure components) relations.

Think of a water wave as a moving car pushing water ahead of it. Just like how the pressure changes in front of the car compared to the back, the pressure at different points under a wave changes based on the wave's dynamics and the water's depth.

So becomes p = 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... 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.

In the modified equation, p represents the total pressure, which is a combination of the dynamic pressure (from the wave motion represented by del phi del t) and the static pressure component (dependent solely on the water depth, represented by -gamma z). The condition p=0 is set to define the boundary at the water surface.

Imagine a swimmer diving underwater. The deeper they go, the greater the pressure they feel from the water above them. The surface pressure can be likened to the static component, while the pressure caused by the wave above represents dynamic pressure.

So p by gamma can be returned as eta K p - z... Now, it has to be mentioned that P was set to 0 to define the free surface boundary condition in Bernoulli’s equation.

This statement introduces the concept of pressure response factor (Kp). Kp takes into account the effects of the wave on the pressure felt at different depths. By expressing pressure as a function of wave height (eta) and applying the boundary conditions, we can assess how pressure responds in relation to wave dynamics. The significance of setting pressure to zero helps establish conditions at the free surface of the water.

Consider throwing a rock into a still pond. The ripples created are like waves, and the pressure exerted at various depths changes based on how high the water is pushed up by the ripples. This resembles the pressure changes described with Kp and can help us understand how waves exert force underwater.

Now, there is something called group celerity, a very important concept. So, when a group of waves or a wave train travels its speed is generally not identical with the speed of the individual waves.

Group celerity is the speed at which a group of waves moves, distinct from the speed of individual waves within that group. This difference in speed is crucial for understanding the overall behavior of waves in fluids, as the collective motion of waves can affect energy transfer and can lead to phenomena such as wave interference.

Think of a parade of floats moving down a street. Each float can move at its own pace (like individual waves), but as a whole, the parade (the group of waves) might move at a different speed. This concept illustrates how group dynamics can lead to differing velocities in water waves.

Thus, the amplitude of the wave and this is the phase and the amplitude varies from 0 to 2 or depending upon the, what the phase is here. So the point of 0 amplitude are called nodes. They can be located by finding the 0s of the cosine factor.

In wave mechanics, the amplitude of a wave, which indicates its height, can change based on the wave's phase. Nodes are points where the amplitude reaches zero, which occur due to the superposition of different waveforms. Understanding where these nodes exist is vital for grasping how wave patterns form.

Consider a jump rope being swung in a wave motion. Points where the rope does not move up or down at all ('zero amplitude') correspond to nodes. This visualization helps in understanding how such points manifest in water waves.

Now, we must go towards wave energy. So, total energy potential energy + kinetic energy... So, the total energy is going to be due to the waves alone is potential energy + kinetic energy, the total energy is gamma a squared by 2.

Wave energy is the sum of its potential energy and kinetic energy, tracking how energy is transmitted through water. The derived equations illustrate that energy is proportional to the square of the wave amplitude, emphasizing that even small increases in wave height can result in substantially more energy. Therefore, in the analysis of waves, understanding energy dynamics is essential.

Imagine a water wheel driven by wave action. The energy transformed to the wheel allows it to turn and do work, similar to how wave energy drives physical processes in marine environments. The total energy can influence activities like electricity generation from ocean waves.

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