Wave properties of light

Light can be modeled as an electromagnetic wave with wavelength, frequency, phase, polarization, and field amplitude. Which values belong in state depends on the question. A timing question may need frequency and phase; a diffraction question needs wavelength and aperture; a polarization question needs two transverse components and their relative phase.

Think like a programmer

Make each unit conversion a named pure function. Store frequency in hertz, wavelength in metres, speed in metres per second, and the medium or refractive-index model that supplied the speed. Do not treat wavelength as an intrinsic photon label that remains fixed across a material boundary: frequency is continuous in simple interface models while phase velocity and wavelength change.

Model checklist

Inputs: Frequency f in Hz, propagation speed v in m/s or a refractive-index model, amplitude, phase, and polarization context.
State: Chosen field components and phase at the current position and time.
Rule: Relate v, λ, and f; evolve phase; preserve frequency across a stationary ideal interface.
Output: Wavelength, field sample, phase difference, and model-specific intensity or polarization description.
Check: v = λf; at fixed f, lowering speed lowers wavelength; phase changes by 2π over one wavelength.

The basic relationship is

\[v=\lambda f,\qquad c=\lambda_0 f,\qquad v=\frac{c}{n}\]

λ₀ is vacuum wavelength, c is vacuum light speed, and n is the refractive index in the simplified nondispersive model. At f = 6 × 10¹⁴ Hz in vacuum, λ = 5 × 10⁻⁷ m, or 500 nm. If the propagation speed is 2 × 10⁸ m/s at the same frequency, the wavelength becomes about 333 nm.

function wavelengthFromSpeed(frequency: number, propagationSpeed: number) {
  if (frequency <= 0 || propagationSpeed <= 0) throw new RangeError("positive finite inputs required");
  return propagationSpeed / frequency;
}

The helper's test fixture uses 6 × 10¹⁴ Hz and 3 × 10⁸ m/s to recover 500 nm. It also checks that reducing speed at fixed frequency reduces wavelength. These direction checks catch common inversions of the equation before they reach a color or ray diagram.

Phase is the repeating coordinate of a sinusoidal field. A difference of 2π represents the same phase, and a phase difference—not simply two amplitude values—controls interference. Polarization describes the orientation and relative phase of transverse field components; it is not a scalar “light strength” setting.

Try this experiment

Prediction: At fixed frequency, entering a higher-index medium reduces wavelength but does not change the ideal interface's frequency.

Calculate wavelength for f = 6 × 10¹⁴ Hz at v = 3 × 10⁸ m/s and 2 × 10⁸ m/s. Predict the ratio before calculating. Then explain which field property a simple color label may hide.

Where this model breaks

The relations here use a single-frequency ideal wave and, unless stated, a nondispersive homogeneous medium. Real media can absorb, scatter, disperse, be anisotropic, or have nonlinear response. A visible-color label is a perception model, not an exact one-to-one encoding of wavelength, and quantum detection needs an additional photon and detector model.

Summary

Wave-property code starts with units and medium assumptions. Use v = λf as a small testable contract, keep phase and polarization as distinct state, and state when dispersion or material response makes the simple conversion inadequate.

Glossary

Wavelength: spatial period of a wave.
Frequency: cycles per second in hertz.
Phase: repeating position within a cycle.
Polarization: orientation and relative phase of transverse field components.
Dispersion: frequency-dependent propagation in a material.

Self-check

What happens to wavelength when speed decreases at fixed frequency?
Which quantity usually remains continuous across an ideal stationary interface?
Why is phase needed to predict interference?

Sources

OpenStax, University Physics Volume 3, The Electromagnetic Spectrum.
E. Hecht, Optics, wave properties and propagation in media.

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 Wave Properties of Light, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Geometric and Wave Optics, 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 Wave Properties of Light 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 Wave Properties of Light into a test

Represent light with explicit wave state, units, and medium-dependent conversion assumptions.

Name the inputs and units that the geometric and wave optics 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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