Energy & Power

Transport in Continuous Media // Lesson 4

The Power of the Invisible

A tsunami in the open ocean might only be 30cm high.

Why does it become a wall of destruction at the shore?

Waves transport **energy**, not mass.
Energy is stored in the **oscillation** of the medium.

Core Question

How do we mathematically quantify the "unseen" energy in a wave?

1. Energy Density

Kinetic Density (\(dK/dx\))

From the motion of the mass elements:

\[ \mathcal{K} = \frac{1}{2} \mu \left( \frac{\partial \psi}{\partial t} \right)^2 \]

Potential Density (\(dU/dx\))

From the stretching of the string:

\[ \mathcal{U} = \frac{1}{2} T \left( \frac{\partial \psi}{\partial x} \right)^2 \]

The Surprising Symmetry

For a traveling harmonic wave, the kinetic and potential energy densities are **exactly equal** at every point in space and time!

\[ \mathcal{K}(x,t) = \mathcal{U}(x,t) \]

2. Transmitted Power

The rate of energy transfer (Power) for a sinusoidal wave:

\[ P_{avg} = \frac{1}{2} \sqrt{\mu T} \omega^2 A^2 \]

\[ P \propto A^2 \]

Amplitude Squared

\[ P \propto \omega^2 \]

Frequency Squared

"To double the power, you only need to increase the amplitude by a factor of \(\sqrt{2}\), OR double the frequency and quarter the amplitude."

3. Wave Intensity

For waves spreading in 3D (like sound or light), we define **Intensity (\(I\))** as Power per unit Area.

\[ I = \frac{P}{4\pi r^2} \]

The Inverse Square Law

Physical Insight

Because \(I \propto A^2\) and \(I \propto 1/r^2\), it follows that for a spherical wave:

\[ A \propto \frac{1}{r} \]

The wave amplitude decays linearly with distance from a point source.

Wave Mechanics Foundations

Discrete to Continuous Waves

Deriving the Wave Equation

Sinusoidal Wave Functions and Phase

Energy and Power Transmission in Waves

The Superposition Principle and Fourier Series Intro

Coupled Oscillator Continuum Worksheet

Discrete to Continuous Waves Slides

Transition Transition Teacher Guide

PDE Derivation Scaffold

Deriving Wave Equation Slides

Wave Equation Answer Key

Sinusoidal Wave Functions Slides

Phase and Partial Derivatives Worksheet

Harmonic Wave Discussion Guide

Wave Energy and Power Slides

Energy and Intensity Worksheet

Energy Transmission Teacher Notes

Superposition and Fourier Slides

Harmonic Synthesis Worksheet

Fourier Demo Guide

Similar Sequences

Molecular Mastery Finals Prep

Olutoyin Makinde

Molecular Chemistry Masterclass

Olutoyin Makinde

Access Tech Lab

Gretchen Ivers

Maîtrise SQL pour Data Analyste

Elhadji Malick DIENE

Maîtrise SQL Détective

Lab Chemistry Mastery

Olutoyin Makinde

The Digital Launch

brittaink@urbandaleschools.com

The Identity Studio

brittaink@urbandaleschools.com

Innovation Lab: Identity Studio

brittaink@urbandaleschools.com

Exposure Mastery

Landry.Stahl

Sticky Matter

Hadlee Richardson

Electron Empire

ssneierson

Square Sage Mastery

mr.sigma

Signal and Story

Tom Ingram

Service Sync

Marlyce Kemp

Efficiency Engine

Matthew.Sprout

Sense Detectives

Michele Magee

Web Interactiva

Norman René Trujillo Zapata

Magnetic Forces Investigation

Olutoyin Makinde

Circuit Mastery

Olutoyin Makinde

Maîtrise ICT Expert

Molecular Frontiers

Tanya Salman

Chemistry Core

Taryn Sigl

Chemical Conversions Mastery

worton

Blueprinting the Air

jmoulison

Retail Career Quest

Tia Washington

Atomic Architecture

Marsha Bolden

Pressure Power

Olutoyin Makinde

Cyber Sentry Prep

Nick Masoner

Body Systems 101

Tanya Salman