2.4. Famous Dispersion Relationship

Now, one important thing after this derivation of the velocity potential is something called a dispersion relationship that is one of the core concept of the wave mechanics. So, the relationship between wavelength with period and water depth is obtained as given below for the dispersion relationship, the main assumption while establishing the relationship is that since we are dealing with small amplitude waves, meaning that the slope of wave profile are so, small that del eta by d eta by dt can be approximately said equal to the vertical component of the velocity w.

In wave mechanics, a critical concept is the dispersion relationship, which relates the wavelength () to the wave period (T) and water depth (d). This relationship is derived under the assumption that we are considering small amplitude waves. Small amplitude means that the height of the waves compared to their wavelength is negligible, resulting in a flat wave slope. Due to this small slope, the vertical velocity (w) can be approximated to the time derivative of the wave elevation (eta). This sets the stage for further relationships in wave behavior.

Think of a gentle wave at the beach that barely changes the water's surface—it's not very steep. This is similar to our small amplitude waves. Just like how the slope of water is minimal for gentle waves, we want to develop relationships based on that minimal change.

So, w can be written as d eta dt or in differential form del eta del t + w is what del eta by del t because that is the vertical velocity correct d eta dt.

With the assumption of small amplitude waves, we express the vertical velocity (w) in terms of the changes in wave elevation (eta) over time (t). By applying the chain rule for differentiation, the change in elevation can be understood as having components related to both time and space. This relationship allows us to combine velocity with spatial changes to derive further important equations.

Imagine tracking how water surface height changes over time as you watch the gentle waves. If you locate a specific point, as the wave passes, both height at that point and speed will change, which is equivalent to how we differentiate eta across different variables.

On canceling the common terms what we get is sigma square by g = k sin h kd divided by cos h k d and this is nothing but tan h kd. So, sigma square can be written as g k into tan h kd. Here sigma is the wave angular frequency equal to 2 pi by T and k = wave number which = 2 pi by L as we have seen before.

After differentiating and simplifying, we arrive at the expression that relates the angular frequency (), gravitational acceleration (g), wave number (k), and hyperbolic functions of the wave parameter (kd). This shows that the square of the angular frequency is proportional to the wave number multiplied by the hyperbolic tangent of k times d. This mathematical formulation is critical for predicting wave behavior in different environments, such as oceans.

This relationship can be likened to how different gears in a vehicle impact speed. Just like how gear changes affect performance based on terrain, this dispersion relationship predicts how waves behave differently based on their depth and wavelength.

Since C = L by T, we get C = we just writing it in a different form. So C can be written as gL by 2 pi tan h kd.

The celerity (C) of a wave is defined as its speed, representing how fast the wave propagates. It's derived from the previous equations linking wavelength (L) and period (T). In the form that relates it to gravitational acceleration and the hyperbolic tangent function, C encapsulates how the wave's speed varies with depth and wave characteristics—a fundamental insight in wave mechanics.

Consider a train moving at different speeds based on the tracks it runs on. Similarly, waves travel at varying speeds depending on their depth—deeper water allows faster movement, akin to how a train would go faster on flattened tracks than on rough terrain.

And this is also a different form of dispersion relationship and because this L appears on both sides of this equation, it has to be solved by trial and error.

The final form of the dispersion relationship encapsulates the intricate interplay between different wave parameters, showing that they influence each other and must often be solved through estimation methods. This encapsulation is not only of academic interest but also critical for practical applications in oceanography and engineering. The relationship informs us about how to estimate wave behavior under various conditions.

Just like a seasoned chef knows how to adjust a recipe based on the ingredients available—sometimes tasting and adjusting as they go—scientists and engineers solve this wave relationship by estimating, testing, and refining their calculations based on various real-life conditions.

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