5.2. Periodic Boundary Conditions

For the waves that are periodic in space and time the boundary condition can be simply represented as, phi as phi of x, t will be same as x + L where L is the wavelength. This is periodic boundary condition in space, the other could be it will be same as at t + delta T.

Periodic boundary conditions are applied when a system exhibits repeating behavior in space and time. For instance, if we have a wave propagating through water, we can define a boundary condition that links the state at one end of a section of the wave to a corresponding state at the other end after a certain distance, represented by the wavelength L. In mathematical terms, this is expressed as phi(x, t) = phi(x + L, t), indicating that the wave's properties at a position 'x' at time 't' are identical to those at position 'x + L' at the same time. The same logic applies to time, allowing us to say that the wave's properties repeat after a period delta T, leading to a similar expression: phi(x, t) = phi(x, t + delta T).

Think of a Ferris wheel at an amusement park. As it rotates, every point at the same height on the wheel repeats its position after completing a full rotation. If you visualize this in terms of time, every moment on the Ferris wheel is similar after a complete rotation. This analogy helps us understand periodic boundary conditions by demonstrating how certain systems exhibit repeating behavior over time and distance.

Now, after reading the boundary condition, we are getting straight to the derivation of the velocity potential to derive the velocity potential there are some assumptions.

In applying periodic boundary conditions, we derive various physical properties of waves, such as velocity potential. To derive the velocity potential, we make specific assumptions about the system. For instance, we often assume that the waves are irrotational, meaning the flow does not possess any vorticity, and that the fluid properties are constant. These assumptions simplify the calculations and allow us to derive elegant mathematical expressions that describe how waves propagate. With the periodic nature of the waves, these derived equations reflect the repetitive patterns observed in real-life wave motion.

Consider a playground swing. The swing moves back and forth in the same manner, creating a repetitive motion every time it returns to a certain point. Similarly, the periodic boundary conditions help us quantify and describe wave motion, akin to capturing the predictable path of the swing in our calculations.

We must find the equation that satisfy that should be satisfied, because of the assumptions we have taken.

When deriving equations based on periodic boundary conditions, it is essential to identify the parameters that must be satisfied by the resulting equations. These include the wave's height, wavelength, and the properties of the fluid through which the wave travels. By considering these aspects, we can create a model that accurately reflects the behavior of the waves under the specified boundary conditions. This involves establishing a mathematical framework that incorporates the continuity and momentum equations to fully characterize the wave dynamics.

Think of baking a cake using a specific recipe. Each ingredient (like flour, sugar, and eggs) must be measured accurately for the cake to turn out correctly. Similarly, in deriving equations for periodic boundary conditions, we must account for each relevant variable in our model to ensure we accurately describe the wave behavior.

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