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A short class with Sarah and her students. The other ways are under More.
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Today, we will explore how water particles move due to wave energy. Can anyone tell me what happens to particles in a wave?
They move in a circular motion?
Great! In fact, their motion is more complicated. We will discover that in shallow water, particles move in elliptical orbits. What does this imply about their horizontal and vertical movements?
It means they have different displacements!
Exactly! And we quantify this with specific equations for horizontal and vertical displacements.
Let's look at the equation for horizontal displacement: . What does represent here?
Is the semi-major axis?
Correct! So then what can we say about the vertical displacement ?
It involves the sine function and indicates the vertical movement of particles.
Well done! Remember, these equations help map out how particles experience wave energy.
Now, let's contrast the two conditions. What happens to particle movement in shallow water compared to deep water?
In shallow water, they move in elliptical orbits.
Exactly! And what about in deep water?
They shift to circular orbits because the equations for displacement align.
Exactly! The critical point to remember is that the semi-major and semi-minor axes become equal in deep water.
Water particles exhibit elliptical motion in shallow water and circular motion in deep water.
The displacement of water particles is expressed through equations with kinematic parameters.
These parameters influence the displacement of waves and their associated movements.
In shallow waters, waves glide, with ellipses on the side.
Imagine a buoy in shallow water; it dances side to side, while deeper, it whirls in circles, taking wave energy for a ride.
Remember 'DEEP' for Circular Orbits: D - Deep water, E - Equal axes, E - Ellipses turn to circles, P - Particles twirl.
CAP - Circular in the deep, Axis equal, Particles spin.
Term
What shape do particles form in shallow water?
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Term
What does the semi-major axis represent?
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The movement of water particles due to wave action, described by equations relating horizontal and vertical displacements.
The longest radius of an elliptical orbit, representing the distance of maximum displacement in the horizontal direction.
The shortest radius of an elliptical orbit, representing the distance of maximum displacement in the vertical direction.
Variables that describe the motion of waves, influencing how water particles displace.
This section covers the water particle displacement in fluid dynamics, detailing the equations governing horizontal and vertical displacements.
Key concepts of water particle displacement are explored through equations that describe horizontal and vertical movements influenced by wave kinematics, leading to the conclusion that water particles exhibit elliptical orbits in shallow water and circular orbits in deep water.
In this section, we discuss the formulae for water particle displacement, emphasizing the role of wave kinematic parameters on both horizontal and vertical motions. The derivations lead us to two critical equations: one for the horizontal displacement (x
given by $$ \frac{\delta x}{D} = \cos(kx - \sigma t) $$ and one for vertical displacement ($$ \delta z = \frac{h}{2} \sin(kd + z) \cdot \sin(kx - \sigma t) $$).
In deeper analyses, we establish that particle displacements in shallow water can be approximated with equations showcasing elliptical orbits, while in deeper water conditions, the trajectories of particles shift to circular orbits due to the equality of semi-major and minor axes. This section concludes with the presentation of the dispersion relationship and its significance in wave mechanics.