Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to Particle Displacement
Unlock Audio Lesson
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.
Understanding Horizontal and Vertical Displacement
Unlock Audio Lesson
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.
Particle Displacement in Shallow and Deep Water
Unlock Audio Lesson
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.
Mathematical Derivation of Displacement
Unlock Audio Lesson
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.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
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.
Detailed
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.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Particle Displacement Expression Overview
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, in this case eta will be a cos k x - sigma t, the u will change u and a x and w and a w, so, if we assume a different velocity potential you remember we had to velocity potentials. So, we started doing all the calculation with the first velocity potential but instead of the first day we started with the second we will obtain this set of the wave kinematic parameters.
Detailed Explanation
In physics, particularly in fluid dynamics, we often need to describe how particles within a fluid move. The term 'eta' relates to the position of a particle over time. Here, the equation 'eta = cos(kx - σt)' shows the oscillatory nature of this motion, where 'k' is the wave number and 'σ' represents the angular frequency. The mention of using different velocity potentials indicates that there are various methods to calculate the displacement of these particles in the fluid.
Examples & Analogies
Imagine standing by the ocean and watching how water moves with the waves. Each wave can be described mathematically, just like how this equation describes particle motion. Just as there are different ways to describe waves (like their height and speed), scientists can use different models (velocity potentials) to understand the same wave motion better.
Water Particle Displacement Calculation
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
the water particle displacement is nothing but integral u times g T w times d T in extend that direction respectively. So, the expression of individual horizontal and vertical particle displacement is integral u dt u we already know in terms of h before.
Detailed Explanation
To find the displacement of water particles, we need to integrate their velocity over time. The expression 'integral u dt' represents this, where 'u' is the velocity. Here, 'h' relates to the height of the wave, linking particle motion to wave properties. Essentially, evaluating this integral helps predict how far a water particle will move vertically or horizontally during a wave's passage.
Examples & Analogies
Think of a child on a swing. If you push the swing gently and then keep track of how far the swing moves over time, you're performing a form of integration. Similarly, calculating the displacement of water particles involves tracking how far they move as waves roll through.
Final Results of Displacement Expressions
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, the final results come out to be h by 2 cos hkd + z phi by sin hkd into cos k x - sigma t. Similarly, the vertical displacement delta z is given by h by 2 sin hkd + z into sin hkd into sin k x - sigma t these are the periodic terms.
Detailed Explanation
The expressions defined provide a mathematical representation of how a water particle moves in relation to both horizontal and vertical displacements. The 'horizontal displacement' equation shows how the wave height (h) and other parameters combine to describe movement. Similarly, the vertical displacement illustrates how waves influence the height of particles in the fluid as they oscillate.
Examples & Analogies
Envision a bobbing buoy in the ocean. As the waves pass, the buoy moves up and down (vertical displacement) and flows along with the wave (horizontal displacement). The equations we derived help predict these movements just like tracking the buoy’s path on a chart.
Elliptical Orbits of Particle Movement
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, what we write is delta x by D whole squared + delta z by B whole squared = 1, this is in general what type of equation if D is not equal to be elliptical. So, this is an equation of analysts showing that water particle moves in in an elliptical orbit.
Detailed Explanation
This equation describes the path of a water particle as an ellipse. Here, 'D' and 'B' represent the semi-major and semi-minor axes, respectively. The equation highlights that the movement of the water particles is periodic and reveals the elliptical nature of their trajectories, which is a fundamental concept in wave mechanics.
Examples & Analogies
Imagine a planet orbiting the sun. Just like planets follow elliptical orbits due to gravitational forces, water particles also move in elliptical paths due to wave energy. This helps visualize complex water movements with simple shapes.
Effects of Water Depth on Displacement
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Now we analyze this displacement in shallow water. So what happens in shallow water for d by L less than 1 by 20 we have 0.05 we have used cos hkd + z and sin hkd + 2 sin hkd + z goes to k d + z and sin hkd goes to kd.
Detailed Explanation
In shallow water, the relationship between depth and wave characteristics alters particle displacement. The equations indicate how wave behavior changes with varying depths. For instance, when considering waves in shallow waters (where depth 'd' is less than a fraction of the wavelength 'L'), the equations simplify, changing how we predict displacement.
Examples & Analogies
Picture standing in a shallow pool versus a deep ocean. In the pool, waves behave very differently—waves might break sooner due to the shallow depth, affecting how much water moves, while in the deep ocean, particles can move more freely without as much interaction with the bottom.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Wave Kinematics: The study of the motion of waves and the forces acting upon them.
-
Displacement Equations: Formulas used to calculate horizontal and vertical displacements of water particles.
-
Orbital Motion: The circular or elliptical path traced by a particle as a wave passes.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
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.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
In shallow waters, we sway and twirl, / Particles move in an elliptical whirl.
📖 Fascinating Stories
-
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.
🧠 Other Memory Gems
-
E to remember: Elliptical in small depths, Circular in deep depths (ECC).
🎯 Super Acronyms
DHB for Displacement, Height, and Depth affecting water particle motion.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Particle Displacement
Definition:
The change in position of a water particle as waves pass through.
-
Term: Wave Height (h)
Definition:
The vertical distance between the crest and trough of a wave.
-
Term: Wave Number (k)
Definition:
A measure of the number of wavelengths per unit distance.
-
Term: Frequency (σ)
Definition:
The number of cycles of a wave that occur in a unit of time.
-
Term: Elliptical Orbit
Definition:
The path followed by water particles in shallow water, resembling an ellipse.
-
Term: Circular Orbit
Definition:
The path followed by water particles in deeper waters, resembling a circle.