Wave speed on a string calculator
Enter the string tension and its linear mass density to get the speed of a transverse wave, v = √(T/μ): 100 N on a string of 0.01 kg/m gives 100 m/s. Made for guitar strings, sonometers and standing-wave labs. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Tension
- 100
- Linear mass density μ
- 0.01
- Wave speed
- 100
- Wave speed unit
- m/s
- Tension over mu
- 10,000
- Model
- ideal_taut_string
A wave speed on a string calculator turns string tension and linear mass density into the propagation speed of a transverse wave. Negative tension or negative density are likewise rejected as outside the physical domain of this calculator.
What wave speed means on a taut string
On an ideal string stretched under constant tension T with uniform linear mass density μ, a small transverse pulse travels along the string at a single characteristic speed given by the closed formula v equals the square root of T divided by μ. The result has SI units of metres per second when T is in newtons and μ is in kilograms per metre.
Physically, higher tension stiffens the restoring force that snaps displaced segments back toward equilibrium, so waves move faster; higher mass per unit length increases inertia and slows the pulse. The formula comes from the one-dimensional wave equation under the usual assumptions: the string is perfectly flexible, tension is large compared with gravity sag, amplitudes are small enough that length changes are second order, and there is no dissipation.
Classroom and lab problems almost always use this ideal model: guitar strings, sonometer wires, and standing-wave apparatus are graded against v equals square root of T over μ before anyone mentions overtones. Frequency and wavelength are related by v equals f lambda once the speed is known, but this calculator returns only the speed itself, not modes or spectra. Teachers care about the zero-density rejection as much as the arithmetic path, because a student who sets μ to zero has left the domain of the continuum string model.
The check path recomputes the ratio and square root with the same fixed rounding used for published fields, so a successful response always ships with steps you can assert in tests or worksheets.
Practical tips, limits, and related searches
Keep tension and linear density inside the published absolute range so the quotient and square root cannot overflow into non-finite results that the solver must reject. Prefer sending numbers rather than pre-formatted strings when you control the client; string numbers are accepted only when they match a strict finite numeric pattern. Use consistent SI units: newtons and kilograms per metre yield metres per second; if your problem states grams per centimetre or dynes, convert first.
Prefer this capability when you already have T and μ under ideal-string assumptions; if your source is a word problem, extract those two numbers and convert units first.
What you can do with it
Homework checks for string wave speed
Students enter tension and linear density from textbook taut-string problems and compare hand-written √(T/μ) against a deterministic result before submitting.
Pre-flight before standing-wave design
Pipelines convert measured tension and mass-per-length into propagation speed first, then branch to wavelength or fundamental-frequency formulas only when v is finite.
FAQ
What is the wave speed on a string?
For an ideal taut string with tension T and linear mass density μ, the transverse wave speed is v = √(T/μ). With T in newtons and μ in kg/m, v is in metres per second.
What happens when linear density is zero?
The formula divides by zero, so the wave speed is undefined.
Does this return frequency or wavelength?
Combine v with f or λ yourself via v = fλ if you need those quantities.
Is the wave speed on a string calculator free?
The browser tool is free and runs locally.
Why does more tension make the wave travel faster?
Higher tension strengthens the restoring force that snaps displaced parts of the string back, so the pulse moves faster. More mass per unit length works the other way: it adds inertia and slows the wave down.
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/phys/wave-on-string-speed \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"tension":100,"linear_density":0.01}'const res = await fetch("https://api.kit.forhosting.com/phys/wave-on-string-speed", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"tension": 100,
"linear_density": 0.01
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/wave-on-string-speed",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"tension": 100,
"linear_density": 0.01
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/wave-on-string-speed", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"tension":100,"linear_density":0.01}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"tension":100,"linear_density":0.01}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/wave-on-string-speed", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"tension": 100,
"linear_density": 0.01
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.wave_on_string_speed",
"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.
Limits
max_abs | 1000000000000 |
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
The model is the classic ideal taut string of uniform mass per unit length under constant tension: small-amplitude transverse waves travel at a single speed independent of frequency. When linear density is zero the formula divides by zero, so the API returns a clear invalid_input error instead of inventing an infinite speed.
How to call the API and what comes back
Tension must be non-negative; linear density must be strictly positive. Values of density whose absolute value is at or below one times ten to the minus fifteen are treated as zero and rejected, matching the product rule that zero μ makes wave speed undefined. Send a JSON body with two required finite numbers: tension (aliases T, force, tension_N, string_tension) and linear_density (aliases mu, μ, mass_per_length, linear_mass_density, density). Absolute values must stay within the published limit of one trillion so intermediate products remain finite in IEEE double precision on the Worker and in the browser. On success the response echoes rounded tension and linear_density, sets wave_speed to v in m/s, wave_speed_unit to the string m/s, tension_over_mu to the T over μ ratio, formula to the classic v equals square root of T over μ string, model to ideal_taut_string, message to a short English summary, and steps to an ordered array of substitution lines. Failed validation—missing fields, non-finite numbers, values outside absolute limits, negative tension, negative density, or zero density—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price.
Practical tips, limits, and related searches
When teaching, pair examples: one hundred newtons and zero point zero one kilograms per metre give v equals one hundred metres per second; quadrupling tension doubles the speed; zero tension yields zero speed at every positive density; and density equals zero demonstrates the hard invalid_input path. This landing targets people searching for a wave speed on a string calculator, tension over linear density, or v equals square root of T over mu online; it is not a full wave-equation solver, not a standing-wave harmonic calculator, and not a medium-dependent bulk sound-speed tool. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners deliberately set density to zero. Round-trip checks are easy: recompute T divided by μ by hand, take the square root, and confirm it equals the published wave_speed within the same twelve-decimal rounding.
Integration use cases
Unit tests for wave and music-physics helpers
Generate known-good speeds at fixed T and μ, zero-tension cases, and deliberate zero-density rejections for coding kata expected values.
Technical questions
Does zero tension mean no wave?
Zero tension yields wave speed zero for any positive density: there is no restoring force to propagate a transverse pulse along the string.