1.2. Linear wave theory

So, linear wave theory, what is that as the name indicates, the linear wave theory can be guessed as you know when the waves are considered linear in nature. So, the background is as everybody knows that the real water waves have you seen waves in the ocean. So, the disturbances that traveling in the ocean are actually waves they might not be linear but they are waves.

Linear wave theory refers to the study of waves under the assumption that they behave in a linear manner. In reality, water waves are complex and may not appear linear at all times, but for theoretical analysis, this simplification allows us to model and understand water wave behaviors more easily. In the ocean, waves travel as disturbances in the surface of the water, which reflects the nature of waves even if they doesn't always appear linear.

Imagine throwing a stone into a calm pond. The resulting ripples spread out in circles, which can initially seem linear if we look closely at small sections of the wave. This linear perspective helps us understand wave dynamics, just like how we can model the growth of a small ripple from a stone drop in a simplified way.

So, the real water waves they propagate in viscous fluid. So, the way the fluid in which it propagates is viscous like water mostly propagates in water.

In the context of wave mechanics, water waves propagate through a viscous medium, which means that the fluid's properties affect how the waves travel. Viscosity describes a fluid's resistance to deformation, which plays a crucial role in how waves interact with the fluid and the boundaries they encounter. Understanding this aspect helps predict how energy from waves dissipates or focuses in certain scenarios.

Think about how oil and water feel different when you stir them. Oil flows more steadily, while water creates ripples easily. In wave mechanics, the presence of viscosity influences how quickly or slowly the wave energy travels through the fluid, similar to how each liquid responds differently when disturbed.

However, we assume that the main body of the fluid is a rotational because the viscous effects but we have learned in the; our laminar and turbulent and viscous fluid flow classes is that the viscosity with the viscosity effects are limited near to the bottom. So,the bottom above that bottom part, the main body of the fluid is in rotational.

In wave theory, an irrotational fluid is one where the flow does not have any rotation or vorticity at large scales. Instead, viscous effects are primarily felt near the boundaries, such as surfaces or bottoms of bodies of water. Within the main volume of the fluid, however, the flow can be assumed to be rotational, allowing for simplified modeling of wave behaviors without considering the complex effects of viscosity across the entire medium.

Imagine a pot of boiling water. Near the surface, the water is moving chaotically with bubbles and currents (viscous effects), but if we look deeper down, the motion is more uniform and chaotic at the surface is less pronounced. This illustrates how, in wave mechanics, we treat the fluid as basically rotational away from its boundaries.

there is a very important concept that is called boundary value problems. This is not only important for this particular chapter, but any problem that has something to do with boundary values for example, we saw in the last week lectures and CFT that boundary values are some of the things that must be specified for the computation to start.

Boundary value problems are critical in wave mechanics as they define the parameters and limits within which the equations governing wave behavior are solved. By specifying certain boundary conditions, we can ensure a unique solution exists for the wave equations being studied, which otherwise might have infinitely many potential solutions.

Consider trying to find the shape of a taut string fixed at both ends. You can stretch it between two points (the boundaries), and based on how you pluck or disturb that string, you can predict the wave forms that arise. The fixed points act as boundary conditions, guiding the behavior of the waves (sound or vibrations) produced.

So, the formulation of boundary value problem is simply the expression of a physical solution in a mathematical forms such that a unique solution exists. So, what happens is if there is an equation that is valid for entire domain entire area. And then there could be infinite number of solutions to that problem.

When approaching boundary value problems, it's necessary to begin by defining the region of interest and formulating the governing equations that apply within that area. By applying specific boundary conditions, we can eliminate infinite possibilities and resolve to a unique solution that describes the physical phenomenon being studied.

Think of a graph with an equation for a curve (like a wave). If you draw that curve without any limitations, you can create infinite variations. However, if you say, 'Let's make this curve touch certain points,' you restrict its variations significantly, allowing for one definitive shape to match your requirements.

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