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A short class with Sarah and her students. The other ways are under More.
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In our previous sessions, we discussed velocity potentials, and today we'll directly analyze how these connect to water particle displacement. Can anyone recall what we mean by velocity potential?
Is it how fast a wave moves through water?
That's a part of it! Velocity potential helps us model the flow of fluids. Now, let’s look at how water particles move under this model. When we find displacement, we see horizontal and vertical components, which are integral u dt and similar terms.
So how do we express these displacements mathematically?
Excellent question! We express horizontal displacement as h/2 cos(hkd) + z multiplied by cos(kx - sigma t). Can anyone note the key terms here?
H is wave height and k is the wave number!
Correct! These are crucial to understanding the motion of our water particles. So, let’s summarize this. The resulting equations enable us to predict how a particle behaves in different water conditions.
Now, let's focus on shallow water conditions. Can someone remind us what happens when the condition d/L is less than 1/20?
The equations simplify, right? D becomes h/2 times a function of kd and z becomes significant.
Absolutely! This occurs because the depth impacts how the water particle's movement manifests. This leads to a more simplified function.
So in shallow water, we're primarily observing elliptical patterns?
Exactly! The elliptical equation helps us visualize the particle's path in this condition, marking its depth and relative amplitude.
Now shifting gears, what can be said about deep-water conditions?
In deep water, do we see circular orbits as D and B become equal?
Spot on! Since both semi-axes converge, the elliptical shape transforms into a perfect circle. This highlights how water particle movement varies with depth.
So we always relate the depth to the type of orbit we see?
Yes! That’s a vital connection. Understanding these properties lets us predict behaviors in wave mechanics.
Formulated to describe the movement of water particles in both elliptical and circular motion.
Differentiation based on ratios of depth versus wavelength impacts particle trajectory.
Understanding motion through mathematical models provides insights into fluid behaviors.
In shallow waters, orbits are wide, water sways in an elliptical glide.
Imagine a fisherman tossing a net wider in shallow waters, casting circular nets in deeper tides.
D and B are key, like the sun and sky, bringing shapes to the waves that fly.
WAVE - Water Analysis and Velocity Evaluation in orbits.
Term
What describes the trajectory of water particles under wave motion?
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How do the parameters D and B relate to particle motion?
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A scalar function used to analyze fluid flow in potential flow theory.
Trajectories of particles in a fluid that can be mathematically described as ellipses.
Half the longest diameter of an ellipse, indicating the extent of horizontal displacement.
Half the shortest diameter of an ellipse, indicating the extent of vertical displacement.
The motion where the path of the particle is a circle, occurring in deep water conditions.
This section explores the equations governing the trajectories of water particles in elliptical orbits, particularly distinguishing between shallow and deep water conditions.
The section details the derivation of the equations for water particle displacement in elliptical orbits, explaining how parameters such as the semi-major and semi-minor axes relate to particle paths in both shallow and deep waters. It also outlines the significance of these equations in understanding wave mechanics.
In the analysis of water particle displacement in waves, we find that the parameters influencing the elliptical orbits are represented by the equations for horizontal and vertical displacements. The particle displacement can be mathematically expressed, and the relationship between the semi-major axis and semi-minor axis is elucidated by the equation delta x/D whole squared + delta z/B whole squared = 1. In shallow water conditions, it is noted that for d/L less than 1/20, the equations transition to a simpler form. Conversely, in deep water, where d/L tends to be larger, the motion is circular as D and B converge. The significance here lies in discerning the changing nature of water particle trajectories based on depth, showcasing how particle movement in water can be classified under elliptical orbits or circular depending on the conditions present.