Energy transport

Waves can move energy through a medium without transporting the medium itself over the same distance. Energy flow depends on both displacement and motion. A moving crest is evidence of phase propagation, not by itself a proof of energy transport; a model needs local stores, a boundary convention, and a transfer/flux diagnostic.

Think like a programmer

Track a local energy estimate beside wave samples. Define the control volume, cell mass/density, tension or stiffness policy, cell width, boundary flux convention, and time staggering. Compute kinetic and deformation energy from the same state arrays used by the wave update. Report total energy, left/right boundary transfer, damping work, and residual at every named output time; the renderer only reads this data.

Model checklist

Inputs: Displacement samples, velocity samples and their time level, tension/stiffness or density model, grid spacing, cell measure, boundary policy, damping/source terms, time step, and energy-flux convention.
State: Per-cell kinetic/deformation energy, total in-domain energy, boundary flux/transfer, damping/source ledger, and residual history.
Rule: Calculate local kinetic energy from motion and potential/deformation energy from spatial differences; sum cells; account for energy entering/leaving through boundaries and declared sources/losses.
Output: Energy-density/flux samples, domain total, boundary-transfer ledger, conservation residual, and time/refinement metadata.
Check: Doubling a small linear-wave amplitude quadruples energy; closed undamped domain energy remains bounded under refinement; outgoing-boundary energy decrease matches recorded flux; damping decrease matches named loss; changing display scale does not change energy arrays.

For a simple taut-string-style discrete model, energy density contains a velocity term and a spatial-slope term. The exact coefficient depends on the physical representation, but the programming pattern is stable: use matching locations and time levels, then compare a domain ledger rather than guessing from brightness or crest speed.

\[\mathcal E\approx\tfrac12\mu(\partial_ty)^2+\tfrac12T(\partial_xy)^2,\qquad \Delta E_{domain}=E_{in}-E_{out}+E_{source}-E_{loss}+r\]

A spatial flux is a signed quantity across a named face. For an open boundary, energy leaving the domain is not unexplained numerical loss; record it with a sign convention. For a reflecting boundary, energy should remain in the domain in the ideal model but can exchange between kinetic and deformation stores. For damping, identify the destination as a reservoir or explicitly modeled loss; simply deleting amplitude makes a conservation claim impossible to audit.

Try this experiment

Prediction: Doubling amplitude raises energy by roughly four in a linear small-amplitude wave model, while an outgoing boundary causes a matched domain-energy decrease and flux increase.

Choose a cell-energy convention and record kinetic, deformation, and total energy for a fixed wave state. Double amplitude without changing grid or stiffness. Then compare reflecting and outgoing boundaries over the same time interval, logging boundary transfer. Add damping and show the separate loss ledger rather than folding it into residual error.

Where this model breaks

Local energy formulas depend on the medium model. Damping and open boundaries should explain expected energy changes. Coarse grids, inconsistent time staggering, nonlinear material response, dispersion, numerical boundary reflections, multidimensional geometry, and unresolved modes can make a simple one-dimensional energy density inaccurate or incomplete.

Wave data sampler

Adjust amplitude and wavelength. The plotted line is a view of sampled displacement data; its speed is frequency divided by wave number.

Amplitude: 1.0Wavelength: 3.0 m

Amplitude 1.0; wavelength 3.0 m; model speed 1.91 m/s.

Try this experiment

Prediction: Doubling amplitude raises energy more than linearly in a linear wave model.

Name the state components needed for kinetic and potential energy at a grid cell.

Where this model breaks

Local energy formulas depend on the medium model. Damping and open boundaries should explain expected energy changes.

Summary

Use local energy and signed boundary-transfer data—not only visible propagation—to reason about transport and loss. Keep the control volume, sampling levels, source/loss terms, and residual visible with every wave claim.

Glossary

Energy flux: energy crossing an area per time.
Kinetic energy: energy of local motion.
Potential energy: stored deformation energy.
Control volume: named domain whose stores and boundary transfers are tracked.
Energy flux: signed transfer rate through a named surface/face.
Self-check
1. Does the medium travel with a wave crest?
2. Which two state features feed a simple local wave-energy estimate?
3. How should an outgoing-boundary energy decrease appear in the ledger?
4. Why is a darker or smaller crest not sufficient evidence of physical loss?

Sources

OpenStax, University Physics Volume 1, wave energy.
M. J. Lighthill, Waves in Fluids.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Energy Transport, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Waves I — Mechanical Waves, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Energy Transport as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Energy Transport into a test

Track local wave energy and flux rather than inferring transport from a moving visual pattern.

Name the inputs and units that the waves i — mechanical waves model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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