2.3. Assumptions of Small Amplitude Wave Theory

So, we apply the change in momentum, we change, rate of change of momentum is the force. So, we obtain half gamma y square b, that is, force on the right hand side minus force on the left hand side is equal to rho b c y mass flow rate.

In this chunk, we discuss the concept of momentum change in the context of fluid dynamics and wave behavior. The change in momentum of fluid is linked to the forces acting on it, which is expressed in terms of the force on both sides of a control volume (e.g., the wave front) and how this relates to the mass flow rate through the area. Specifically, half the density of the fluid multiplied by the square of the water depth gives the force balance, leading to an expression that incorporates mass flow into the relationship.

Think of a crowded bus: every time someone gets on or off, the momentum of the bus changes slightly. The same concept applies in fluid dynamics; when water moves through a channel, the way it speeds up or slows down changes depending on how many 'people' (or particles) are moving through.

So, if you assume that delta y square, so, delta y is small amplitude wave theory is already assumed. So, delta y is very less compared to y.

This chunk delves into the assumption of small amplitude in wave theory. It states that the changes in wave height (delta y) are negligible compared to the overall height of the water (y). This assumption simplifies mathematical expressions and allows us to derive important relationships, because it implies that certain terms can be ignored when they become extremely small, making calculations more manageable.

Imagine a small ripple compared to a large lake; the ripple's height is tiny compared to the vastness of the lake itself. This analogy demonstrates why we can ignore smaller variations when studying overall wave behavior.

So, we just look at this term, and actually if we take half, half common, gamma, gamma common and this b, b common. So, the left hand side I am writing only just trying to, you know, half gamma, b is going to be common, it will be y square - y + delta y whole square ... So, gamma is rho g, so, we can write, rho g delta y is equal to rho c delta v.

In this chunk, the derivation of the wave speed equation is elaborated. By applying the assumptions and equation manipulations, we reach a point where we relate changes in wave height and velocity to gravity. By simplifying, we uncover that the wave speed 'c' can be expressed in terms of gravitational acceleration (g) and the water depth (y). This mathematical process reveals significant insights into how waves behave dynamically.

It's similar to how a kid on a swing accelerates downwards due to gravity. The deeper you push into the swing (like increasing depth), the faster the swing will go. In waves, if the water is deeper, the waves can travel faster as well!

So, fluid density is not an important parameter. And why is that? ... Therefore, rho and rho gets cancelled out from both side and that is why in fluid density is not an important parameter in this wave of the speed of the wave.

This chunk talks about an interesting aspect of wave speed, specifically that fluid density does not affect the speed of small amplitude waves. This occurs because both inertial and hydrostatic pressures depend on density, cancelling each other out in the wave speed equations. Thus, it shows that for small amplitude waves, the speed is solely determined by gravitational effects and water depth.

Consider a pool where people of different sizes jump into the water. Whether it's a child or an adult, if they both jump in the same way, the ripples they create will follow similar patterns in the water. Here, the weight (analogous to density) matters less than how you jump (akin to gravity).

So, if we consider a fluid which is flowing to the left with speed v ... the relative velocity will be c - V for the waves.

In this portion, the relationship between wave speed and fluid velocity is presented, introducing the concept of the Froude number. This helps classify flow regimes into subcritical or supercritical based on the relationship between fluid speed (v) and wave speed (c). If the wave speed is greater than the fluid speed, waves can travel upstream, which defines subcritical flow; if not, they cannot, marking supercritical flow.

It's like a boat in a river. If the boat moves faster than the water current, it can go upstream. But if the river’s current is stronger than the boat, it will float downstream regardless. Understanding how waves interact with flow speeds is crucial in any fluid system.

Now, this if you see this equation, this will means if the amplitude delta y is larger, c will be larger. ... the waves which has larger amplitude will travel faster.

This part emphasizes a critical departure from the small amplitude theory: once the waves reach a finite size, their speed starts to depend on their amplitude. Larger waves propagate faster, which is fundamentally different from the initial assumption that speed remains constant regardless of amplitude. This understanding aids in predicting the behavior of larger waves, such as tsunamis or storm surges.

Think about a trampoline: the higher you jump (greater amplitude), the faster you come down and bounce back up. Larger waves behaving similarly tell us more about their dynamics and potential impacts.

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