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Today, we’re diving into the expression of particle displacement in waves. Can anyone tell me what we mean by particle displacement?
Is it how far a water particle moves from its original position?
Exactly! Particle displacement refers to the change in position of a water particle as waves pass through. Let’s denote this horizontally as Δx and vertically as Δz.
How do we express these displacements mathematically?
Great question! The horizontal displacement can be expressed as Δx = d cos(kx - σt), where d is the wave amplitude, and the vertical displacement as Δz = B sin(kx - σt). Let’s explore what these components mean.
We have two equations for displacement. In terms of parameters meaningful in wave motion, we find horizontal and vertical displacements using the wave height and mathematical functions. Can someone recall what parameters are essential in these expressions?
I think it's the wave height, wave number, and frequency.
Correct! The wave height h, wave number k, and frequency σ play critical roles. The displacement in shallow waters, as expressed, forms an elliptical path, represented by the equation Δx²/D² + Δz²/B² = 1.
And what happens in deeper waters?
In deeper waters, the parameters change such that both D and B equal h/2 e^(kz), leading to circular orbits described by the equation.
Now, let’s analyze how particle displacement varies. In shallow waters, the depth-to-wavelength ratio impacts how particles move. What do we know about this condition?
When d/L is less than 1/20, the motion is elliptical?
Exactly! This results in a significant downward rotation. In contrast, in deep water, d/L exceeds this ratio, causing the particles to follow circular orbits.
So, can we visualize how these orbits look?
Certainly! In shallow water, they appear like ellipses, while they become circles in deep water, leading to different behaviors under varying conditions.
Let’s derive equations for horizontal displacement and vertical displacement once more. Why do we perform these derivations?
To establish a quantitative measure for how particles move with wave propagation.
Exactly! We express Δx and Δz mathematically to find relationships and understand particle behavior in waves. This also prepares us for future, more complex calculations!
What about when we use MATLAB for deeper computations?
Good point! MATLAB can perform numerically intensive tasks and utilize derived equations effectively, simplifying complex calculations for engineers and scientists.
Example 1: A wave with a height of 2 meters shows a particle displacement in shallow water that results in elliptical orbits with notable depth effects.
Example 2: In deeper waters, wave particle displacement results in circular orbits, simplifying calculations for marine engineering applications.
In shallow waters, we sway and twirl, / Particles move in an elliptical whirl.
Imagine a water particle beginning at rest. As waves approach, it starts moving in patterns, first in an ellipse in shallow water, then perfectly circular in deep water, adapting with the depth.
E to remember: Elliptical in small depths, Circular in deep depths (ECC).
DHB for Displacement, Height, and Depth affecting water particle motion.
Term
Particle Displacement
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Term
Shallow Water Displacement
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Term
Deep Water Displacement
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The change in position of a water particle as waves pass through.
The vertical distance between the crest and trough of a wave.
A measure of the number of wavelengths per unit distance.
The number of cycles of a wave that occur in a unit of time.
The path followed by water particles in shallow water, resembling an ellipse.
The path followed by water particles in deeper waters, resembling a circle.
This section discusses the mathematical expressions for water particle displacement in wave motion, analyzing horizontal and vertical components based on wave kinematics.
In this section, the derivation of equations for horizontal and vertical particle displacement in wave motion is presented. By examining the effects of different wave parameters, it reveals how particles move in elliptical orbits in shallow waters and circular orbits in deeper waters.
In this section, we explore the formulas for water particle displacement due to wave motion. It starts with the basic definitions of velocity potential and kinematic parameters, highlighting how displacement can be computed in both shallow and deep water scenarios. The section introduces the mathematical expressions for horizontal and vertical displacements, represented as functions of wave properties including wave height and depth. As we derive these expressions, we observe that in shallow water, particle displacement is elliptical, while in deep water, it transitions to circular orbits. The role of wave height and wavelength in defining these displacements is emphasized, culminating in a discourse on analytical orbits in varying depths. Overall, the section provides crucial insights into wave mechanics and their implications for understanding fluid dynamics.