Wave amplitude reflection coefficient calculator
See how strongly a wave bounces back where the impedance changes: from 1.5 to 4.5 the reflection coefficient is 0.5, so a quarter of the power reflects and 0.75 passes on. Works for sound, cables and strings. Free in your browser.
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- Your data stays on your device
Run — free
Worked example
- Z1
- 1.5
- Z2
- 4.5
- Impedance ratio
- 3
- Reflection coefficient
- 0.5
- Amplitude reflection
- 0.5
- Transmission coefficient
- 1.5
The amplitude reflection coefficient at a planar interface tells you how much of an incident wave reflects when it meets a change in characteristic impedance. For normal incidence the standard formula is R equals open parenthesis Z2 minus Z1 close parenthesis over open parenthesis Z2 plus Z1 close parenthesis, where Z1 is the impedance of the medium carrying the incident wave and Z2 is the impedance of the second medium. The same algebra covers acoustic pressure waves with acoustic impedance in rayls and electromagnetic or transmission-line voltage waves with characteristic impedance in ohms.
What the amplitude reflection coefficient means and which two impedances you need
When a wave travels from medium one into medium two, continuity of the appropriate field quantities at a lossless planar boundary fixes the ratio of reflected to incident amplitude once the two characteristic impedances are known. That ratio is the amplitude reflection coefficient R equals (Z2 minus Z1) divided by (Z2 plus Z1). If Z2 is greater than Z1, R is positive and the reflected pressure or voltage wave keeps the same sign as the incident wave.
If Z2 is less than Z1, R is negative and the reflected wave flips sign, a pi phase inversion that textbooks illustrate with a free end versus a fixed end on a string or with a soft acoustic boundary. If Z2 equals Z1 the media are matched, R is zero, and there is no reflection. Search intent clusters around phrases such as amplitude reflection coefficient calculator, acoustic impedance reflection, characteristic impedance reflection coefficient, wave reflection R equals Z2 minus Z1 over Z2 plus Z1, and intensity reflection R squared.
People want numeric R and T they can check against a worksheet or embed in a pipeline, not only a reminder that the formula exists. Keep both impedances in the same physical unit; the engine is unit-agnostic and does not convert rayls to ohms or model complex, frequency-dependent Z.
How this calculator validates input and applies R = (Z2 − Z1) / (Z2 + Z1)
Numeric strings that parse to finite positives are accepted so forms and CSV-style pipelines can pass values without an extra cast. Provide only one name per impedance group; sending both z1 and Z1 at once is rejected so the active value is unambiguous. After validation the denominator D equals Z2 plus Z1 is formed, R equals (Z2 minus Z1) over D, and the pressure-style amplitude transmission T equals two times Z2 over D.
Intensity reflection for real positive impedances is R squared; intensity transmission is returned both as one minus R squared and as the equivalent form four times Z1 times Z2 over D squared so you can cross-check homework identities. Classic classroom checks include a soft-to-hard step with Z2 much larger than Z1 (R near plus one), a hard-to-soft step with Z2 much smaller than Z1 (R near minus one), and a matched pair with equal Z (R exactly zero).
There is no multilayer stack, no absorption, no complex Z, and no angle of incidence other than normal; this is ideal single-interface algebra from the two known impedances.
What you can do with it
Acoustics and ultrasonics homework
Students and graders compute R and T for soft-to-hard and hard-to-soft impedance steps, check the sign of R, and confirm intensity reflection equals R squared.
Probe and tissue interface checks
Quick verification of amplitude and intensity reflection at an ultrasound or audio boundary from two acoustic impedances before deeper layered models.
RF and transmission-line mismatch sketches
Notebooks and lesson platforms pin deterministic R from real characteristic impedances Z1 and Z2 without embedding Smith-chart code for a single real step.
FAQ
What happens if an impedance is zero or negative?
Characteristic and acoustic impedances must be finite and strictly positive; validation failures are not charged.
Is the wave reflection coefficient calculator free?
The browser tool runs free on your device.
Does this compute SWR or handle complex impedances?
No. This capacity solves normal-incidence amplitude R and T from two real positive impedances. For VSWR from a known |Γ|, use a dedicated SWR-from-reflection tool; complex Z is out of scope here.
What does a negative reflection coefficient mean?
The second medium has a lower impedance than the first, so the reflected wave comes back with its sign flipped. That is the half-cycle phase inversion textbooks show with the loose, unfixed end of a string.
How much power is reflected when the impedance triples?
Going from 1.5 to 4.5 gives a reflection coefficient of 0.5. Reflected power is that value squared, 0.25, so 0.75 of the incident power continues into the second medium.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/optics/wave-reflection-coefficient \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"z1":1.5,"z2":4.5}'const res = await fetch("https://api.kit.forhosting.com/optics/wave-reflection-coefficient", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"z1": 1.5,
"z2": 4.5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/optics/wave-reflection-coefficient",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"z1": 1.5,
"z2": 4.5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/optics/wave-reflection-coefficient", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"z1":1.5,"z2":4.5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"z1":1.5,"z2":4.5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/optics/wave-reflection-coefficient", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"z1": 1.5,
"z2": 4.5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "optics.wave_reflection_coefficient",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Students checking homework on soft-versus-hard boundaries, ultrasonics engineers sizing probes against tissue layers, RF designers estimating a simple real-load mismatch, and tutoring platforms that need structured JSON all need a deterministic reflection-coefficient calculator rather than a hand-wavy diagram. This capacity accepts two positive finite impedances, returns R, T, intensity reflection and transmission, a phase note for the sign of R, and a short expression string, and rejects zero, negative, or non-finite values with a clear invalid_input error.
What the amplitude reflection coefficient means and which two impedances you need
Delivering z1, z2, impedance_ratio, reflection_coefficient, transmission_coefficient, power_reflection_coefficient, intensity_transmission, phase_note, formula, method, and expression as JSON makes the result easy to paste into notes, diff in automated tests, or reuse across languages without reimplementing the algebra.
How this calculator validates input and applies R = (Z2 − Z1) / (Z2 + Z1)
Send a JSON body with required z1 and z2. Aliases Z1, impedance1, incident_impedance, z_incident, and Z_i map to the incident-medium impedance; aliases Z2, impedance2, transmitted_impedance, z_transmitted, and Z_t map to the second medium. Every provided field must be a finite positive number: missing fields, empty strings, non-numeric text, zero, negative values, NaN, and Infinity all fail immediately with invalid_input before any result is returned, and failed validation is not charged. Results are rounded to a stable twelve-significant-digit precision so the same input always yields the same JSON across JavaScript engines.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate reflection-coefficient check while grading acoustics homework, verifying an ultrasound soft-tissue interface sketch, comparing a free-end versus fixed-end sign convention, or sanity-checking a real characteristic-impedance step on a transmission line before deeper Smith-chart work. Always put the incident medium first as z1 so the sign of R matches the usual textbook convention; swapping labels flips the sign of R and changes T, which is correct physics for the reversed interface direction, not a bug. There is no language model, no external acoustics or RF service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. Each successful request is one billable item at $0.002; failed validation does not consume balance. Results are fully deterministic: the same two impedances always yield the same reflection_coefficient and transmission_coefficient. Wire the endpoint into wave-physics drills, contest answer checkers, probe-matching worksheets, or preprocessing steps where you already prefer explicit domain errors—such as rejecting a non-positive impedance—over opaque black-box widgets that cannot explain why a bad input was refused. For voltage standing-wave ratio from a known reflection magnitude, use a dedicated SWR capacity instead of this one; this endpoint answers the impedance-to-R question, not the R-to-SWR question.
Technical questions
How do I calculate the amplitude reflection coefficient from two impedances?
Send JSON with z1 (incident medium) and z2 (second medium), both finite and positive. The response includes reflection_coefficient R=(Z2−Z1)/(Z2+Z1), transmission_coefficient T=2·Z2/(Z2+Z1), and intensity terms.
What does a negative reflection coefficient mean?
R < 0 when Z2 < Z1: the reflected pressure or voltage wave is inverted by π relative to the incident wave. phase_note reports negative_R_pi_phase_inversion in that case.