2.6. Froude Number Effect on Solitary Waves

So, actually, I have written down the reason, it was not supposed to, but it is okay. We will I will tell you; the reason is the wave motion is a balance between the inertial effects, which is proportional to rho and also the hydrostatic pressure effect, which is proportional to rho g.

The Froude number is a dimensionless number that is used to compare inertial forces to gravitational forces in fluid dynamics. It is defined as V/sqrt(g*y), where V is fluid velocity, g is gravity, and y is the flow depth. It helps us understand whether the flow is subcritical (Froude number < 1) or supercritical (Froude number > 1). In terms of wave motion, the relationship indicates that inertial forces (which depend on density) and hydrostatic pressure forces are in balance.

You can think of the Froude number like a seesaw at a playground. One side represents the weight of gravitational forces (like the kids sitting on one end), while the other side represents inertial forces (like the kids on the other end). For balance, the see-saw needs to have the right ratio of weight on each side, similar to how fluid motion is balanced between these forces.

So, the Froude number for solitary wave is V / c, where c is the speed of the wave and V is the velocity of the fluid with the water with the velocity of the fluid, which will be definitely different from the waves speed.

The Froude number for solitary waves is calculated by dividing the velocity of the fluid (V) by the wave speed (c). This ratio helps predict how the wave will behave in relation to the flow of the fluid. If the wave speed is greater than the fluid speed (c > V), the waves can travel upstream, indicating a subcritical flow. Conversely, if the fluid speed exceeds the wave speed (V > c), the waves cannot travel upstream, indicating a supercritical flow.

Imagine a boat on a river. If the current of the river (V) is slower than the speed of the boat (c), the boat can sail upstream against the current. However, if the river current is faster than the boat, it’s swept downstream regardless of the boat’s ability. Similarly, the Froude number tells us about the relative strengths of waves and the fluid currents.

If c is greater than V, so what is going to be the Froude number, less than 1, so that means subcritical flow, very simple to see. If c is less than V, so that means V is greater than c waves do not travel upstream and because of this definition V / c, because V is greater than c Froude number is greater than one and this falls into regime of supercritical flow.

In fluid dynamics, flow can be classified into subcritical and supercritical based on the Froude number. A Froude number less than 1 indicates subcritical flow, where the wave speed (c) is greater than the fluid speed (V), allowing waves to propagate upstream. When the Froude number is greater than 1, it indicates supercritical flow, where the fluid speed exceeds the wave speed, and waves cannot travel upstream. This classification is crucial for understanding wave behavior in channels.

Consider a water slide at a park. When the slide is steep and fast (supercritical flow), if you try to roll back up by swimming, you won’t succeed because the slope is too steep (fluid speed greater than wave speed). But on a gentle slope (subcritical flow), you have a chance to paddle back upstream since the slide isn't pushing you down as fast.

But if the waves are of finite size that means, they are no longer small amplitude, no longer small amplitude, then the wave speed actually is given by this, you can try to remember this formula, but, the derivation is not in the scope of this course. So, c is given by under the root of g y multiplied by one + delta y / y to the power half. And this is equation number 5.

For small amplitude waves, the speed is independent of amplitude. However, when the waves become of finite size, the speed depends on their amplitude. The new formula indicates that as the amplitude increases, the wave speed also increases. This adjustment factor (delta y / y) accounts for the effect of wave amplitude on speed, showing distinctive behavior as the wave scales change.

Think of a child on a swing. When the swing is pushed gently (small amplitude), it swings at a certain speed. But if the swing is pushed with more energy (larger amplitude), it swings faster. Similarly, for waves, smaller amplitudes result in consistent speed, but larger amplitudes lead to faster wave propagation.

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