Standing wave antinodes count calculator
The third harmonic of a string fixed at both ends has 3 antinodes, 4 nodes and 3 loops: enter any harmonic number and get the antinode count, which always equals that number, without drawing the mode. Free in your browser.
- Free
- Runs in your browser
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- Your data stays on your device
Run — free
Worked example
- Harmonic
- 3
- Antinodes count
- 3
- Node count
- 4
- Loop count
- 3
- Ends
- fixed-fixed
Standing waves on a string fixed at both ends appear in every introductory wave-physics unit, music-acoustics lab, and engineering vibration course. Students often need a fast answer to a narrow question: how many antinodes does the nth harmonic contain? For fixed-fixed boundary conditions the answer is exact and integer: the antinode count equals the harmonic number itself.
Why fixed-fixed antinode count equals the harmonic number
A standing wave forms when two waves of equal frequency and amplitude travel in opposite directions and interfere. On a string fixed at both ends, displacement must stay zero at the supports, so those ends are always nodes. Between the ends the string divides into equal half-wavelength segments.
The fundamental mode (n equals one) holds one half-wavelength: a single antinode sits in the middle and two nodes sit at the ends. The second harmonic (n equals two) holds two half-wavelengths: two antinodes and three nodes. In general the wavelength satisfies lambda equals two L over n for string length L, antinodes appear at odd multiples of L over two n, and there are exactly n of them.
Related counts follow at once: nodes total n plus one (the n minus one interior nodes plus the two ends), and loops equal n. Open-open pipes and free-free rods use different boundary conditions and different count rules; this calculator is intentionally scoped to fixed-fixed strings so the answer stays unambiguous for exam language that says both ends fixed.
Where an antinode count comes in handy
Classrooms and tutoring platforms use a standing wave antinodes count calculator so learners can check the integer answer for arbitrary harmonic numbers without sketching every mode by hand. Exam-prep generators need a deterministic oracle that marks wrong a claim that the fifth harmonic has four antinodes, and that rejects fractional n instead of flooring silently. Music and instrument-design notebooks often map fretted-string partials to mode indices and want the antinode count for visualisation or sensor placement without importing a full wave simulator.
Failed validation does not consume balance. Keep requests small, pin the integer fields in tests, and you get reproducible counts for homework, LMS plugins, and edge workers without standing up symbolic physics infrastructure of your own.
What you can do with it
Exam and LMS auto-grading
Mark standing-wave antinode questions with a deterministic oracle that rejects non-integer or non-positive modes instead of inventing a count.
Lab scripts and mode visualisers
Drive string-mode diagrams or sensor-placement notes from the integer antinode count without computing full positions when length is unknown.
FAQ
Do I need the string length L?
No. Antinode count under fixed-fixed ends depends only on the mode index n. Use a geometry calculator if you need wavelength or positions that involve L.
What happens if harmonic is zero, negative, or not an integer?
Only positive integers n ≥ 1 are valid mode indices for this count.
Does this work for open pipes or free ends?
Open-open or closed-open pipes use different antinode rules relative to n.
How many antinodes does the third harmonic have?
Three. On a string fixed at both ends the antinode count always equals the harmonic number, so the third harmonic shows 3 antinodes, 4 nodes counting the ends, and 3 loops between them.
Does it also give the number of nodes and loops?
Yes. Along with the antinodes you get the node count, which is the harmonic number plus one because both fixed ends are nodes, and the loop count, which equals the harmonic number.
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/standing-wave-antinodes-count \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"harmonic":3}'const res = await fetch("https://api.kit.forhosting.com/optics/standing-wave-antinodes-count", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"harmonic": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/optics/standing-wave-antinodes-count",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"harmonic": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/optics/standing-wave-antinodes-count", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"harmonic":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"harmonic":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/optics/standing-wave-antinodes-count", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"harmonic": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "optics.standing_wave_antinodes_count",
"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
This standing wave antinodes count calculator accepts a positive integer harmonic n, returns the antinode count together with the matching node and loop counts, and rejects non-integer or non-positive modes with a clear invalid_input error instead of a silent wrong integer.
Why fixed-fixed antinode count equals the harmonic number
That identity—antinodes_count equals n—is the contract of this endpoint. It does not need L because the count is pure topology of the mode shape under fixed-fixed ends.
Input contract, response fields, and error behaviour
Values above ten thousand are rejected as out of practical classroom and lab range so a typo cannot request absurd mode indices. Example: harmonic equals three yields three antinodes, four nodes, and three loops—the classic third-harmonic picture with three peaks between fixed ends. POST a JSON body with one required field: harmonic, a positive integer mode index. Aliases n, mode, harmonic_number, and nth are accepted so clients can reuse names from lab sheets or sibling standing-wave tools. The value may arrive as a JSON number or as a numeric string; exact integer floats such as three point zero are accepted after rounding within a tiny epsilon. Missing fields, empty strings, NaN, Infinity, zero, negatives, and true non-integers fail with invalid_input naming the problem, and those failures are not billed. On success the body echoes harmonic as a clean integer, returns antinodes_count equal to that integer, node_count equal to n plus one, loop_count equal to n, ends set to the fixed-fixed tag, plus short formula and relation strings you can display next to the number. Pair this count API with a geometry tool when you also need positions or wavelength; keep this call when the only question is how many antinodes the mode carries.
Integration use cases
Classroom and homework checks
Let students submit harmonic n for a fixed-fixed string and verify that antinodes_count equals n, with node_count n+1 and loop_count n.
Technical questions
How many antinodes does the nth harmonic have on a fixed-fixed string?
Exactly n. Send a positive integer harmonic; the API returns antinodes_count equal to that harmonic number, plus node_count = n+1 and loop_count = n.
What does the standing wave antinodes count calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.