Calculate wave transmission coefficient from amplitude, flux, normalization, or reflection, or solve amplitude and lossless reflection from a known coefficient.

Use T = |t|² only when incident and transmitted amplitudes have equal flux normalization.

Nonnegative magnitude of transmitted/incident amplitude; not a flux percentage.

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Related Calculators

Transmission Coefficient Formula

The transmission coefficient measures how much of an incident wave successfully passes through a boundary, interface, or potential region. In the most general form, it is defined as transmitted flux divided by incident flux. In many introductory problems, the transmitted and incident waves use the same normalization, so the relation simplifies to the square of the transmission amplitude magnitude.

T = jtransmitted / jincident

T = (Ftransmitted / Fincident) × |t|²

If Ftransmitted = Fincident, then T = |t|²

Select matched or unequal normalization, direct flux ratio, or a lossless reflection complement. For unequal normalization, define flux magnitude as j = F × |amplitude|² on each side. Enter positive factors in consistent units for the actual wave variable. They are not universally wave velocities: quantum particle current, optical field amplitudes and acoustic pressure amplitudes have different flux relations.

How to Use the Calculator

  1. Select the quantity and calculation basis from Solve for.
  2. Enter all displayed inputs. For flux ratios, use matching flux units; for unequal normalization, use the same convention for both positive factors.
  3. Select Calculate. Enter dimensionless fractions, such as 0.64 for 64%.
  4. Review the result, assumptions and calculation details. Select Calculate again after an edit.

Reverse Formula

If you already know the transmission coefficient and need the amplitude magnitude in the matched case, take the square root:

Matched normalization: |t| = √T

Variable Definitions

  • T = transmission coefficient, a dimensionless ratio of transmitted flux to incident flux
  • t = transmission amplitude, generally complex
  • |t| = magnitude of the transmission amplitude
  • jtrans = transmitted flux
  • jinc = incident flux
  • Ftransmitted, Fincident = positive real flux-per-squared-amplitude factors for the chosen wave variable, with consistent units. Propagating quantum particle waves can use velocity; other conventions may involve impedance, admittance, refractive index and angle.

What the Result Means

  • T = 0: no transmission
  • 0 < T < 1: partial transmission
  • T = 1: complete transmission

For a correctly normalized passive system, transmitted flux fraction T lies from 0 to 1. An amplitude magnitude greater than 1 can still be passive when its flux normalization differs. For example, |t| = 2 with Fincident = 4 and Ftransmitted = 1 gives T = 1. Above-unity T is retained as a diagnostic; check normalization, measurement boundaries or gain.

Examples

Example 1: Find transmission coefficient from amplitude magnitude

If the transmission amplitude magnitude is 0.8:

T = |t|² = 0.8² = 0.64

This means 64% of the normalized incident flux is transmitted.

Example 2: Find amplitude magnitude from transmission coefficient

If the transmission coefficient is 0.25:

|t| = √0.25 = 0.5

The corresponding transmission amplitude magnitude is 0.5.

When the Simplified Formula Is Valid

The shortcut T = |t|^2 is valid when the incident and transmitted waves are normalized the same way. That often happens in simplified one-dimensional scattering setups where the wave speeds or wave-number factors on both sides are equal. If they are not equal, the flux ratio must include the extra velocity or normalization term.

T generally differs from |t|² unless the flux normalization factors match

Transmission vs. Reflection

Transmission is commonly analyzed together with reflection. For a lossless two-channel system, the transmitted and reflected fractions sum to 1:

R + T = 1, for a lossless two-channel model

Here, R is the reflection coefficient. If transmission increases, reflection usually decreases, assuming no absorption or gain.

Common Applications

  • Quantum mechanics and tunneling problems
  • Wave propagation across boundaries
  • Optics and layered media
  • Acoustics and impedance transitions
  • Electromagnetic wave transmission
  • General scattering and interface analysis

Common Mistakes

  • Using T = |t|^2 when the incident and transmitted waves are not equally normalized
  • Confusing the amplitude t with the coefficient T
  • Forgetting that t may be complex while the calculator uses only |t|
  • Assuming all systems must satisfy R + T = 1 even when absorption or gain is present
  • Entering percentages instead of decimal form, such as using 64 instead of 0.64

Quick Reference

Known Value Use This Relation Result
Amplitude magnitude |t|

Matched normalization: T = |t|²

 Transmission coefficient
Transmission coefficient T

Matched normalization: |t| = √T

 Amplitude magnitude
Unequal normalization

T = (Ftransmitted / Fincident) × |t|²

 Flux-corrected transmission

Frequently Asked Questions

Is the transmission coefficient the same as the transmission amplitude?
No. The amplitude is usually a complex quantity, while the transmission coefficient is a real, dimensionless flux ratio. In the matched case, the coefficient equals the square of the amplitude magnitude.

Can the transmission coefficient be greater than 1?
In standard passive normalized systems, it is usually between 0 and 1. Values outside that range typically indicate a different normalization, gain, or a modeling issue.

Why does the calculator ask for amplitude magnitude instead of amplitude?
Because the matched-case relation depends on |t|, not the phase of the complex amplitude.

What if I know transmitted and incident flux directly?
Then compute the coefficient from the flux ratio:

T = jtransmitted / jincident

Summary of the Calculation

The calculator converts between amplitude magnitude and flux transmission under matched or explicitly supplied unequal normalization, computes direct flux ratios, and uses R + T = 1 only in lossless two-channel modes. It does not recover complex phase or solve an interface, tunneling barrier, absorption model, evanescent channel or complex medium. Confirm the relevant wave-specific flux relation before interpreting a result.