ForHosting KIT · Science

Group velocity calculator

Turn two measured dispersion points into the speed of a pulse envelope: angular frequencies of 1e15 and 1.2e15 rad/s at wave numbers of 1e7 and 1.1e7 rad/m give 2e8 m/s. Free in your browser.

  • Free
  • Runs in your browser
  • No sign-up
  • Your data stays on your device
Run — free How it works & API
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Run — free

Worked example

If you enter
Omega1
1,000,000,000,000,000
K1
10,000,000
Omega2
1,200,000,000,000,000
K2
11,000,000
You get
Delta omega
200,000,000,000,000
Delta k
1,000,000
Group velocity
200,000,000
Vg
200,000,000

Group velocity is the speed at which the envelope of a wave packet—and therefore the energy and information it carries—travels through a dispersive medium. For a continuous dispersion relation ω(k), the textbook definition is the derivative vg equals dω over dk evaluated near the carrier wave number. In lab notes, spectroscopy tables, and numerical simulations you rarely hold a closed-form ω(k); you hold two nearby points and need a trustworthy finite-difference estimate. This group velocity calculator accepts angular frequencies ω1 and ω2 together with wave numbers k1 and k2, forms Δω and Δk, and returns vg equals Δω over Δk with stable rounding, display equation, and step strings. Equal wave numbers are rejected because the denominator would vanish.

What group velocity means and how the finite difference works

A monochromatic plane wave travels at the phase velocity vp equals ω over k, but a localized pulse is a superposition of many wave numbers. When the medium is dispersive, different spectral components advance at different phase speeds and the packet envelope moves at the group velocity. In the slowly varying envelope approximation that envelope speed is exactly the derivative of angular frequency with respect to wave number: vg equals dω over dk at the carrier.

When you only have tabulated or measured pairs (k, ω), the simplest unbiased local estimate is a centered or one-sided first difference between two samples. This tool implements that estimate as vg equals (ω2 minus ω1) divided by (k2 minus k1). The spacing |Δk| should be small enough that the local slope is nearly constant, yet large enough that floating-point noise and measurement uncertainty do not dominate the quotient—standard numerical-differentiation practice.

Units are your responsibility: with ω in radians per second and k in radians per meter, vg is meters per second; any other consistent pair (for example optical frequency in rad/fs and k in rad/μm) still yields a consistent speed in the matching unit system. The capability does not fit a full dispersion model, does not correct for material absorption, and does not convert between frequency f and angular frequency ω—you must supply angular quantities already.

Phase velocity is intentionally omitted so the page stays focused on the search intent group velocity calculator and the finite-difference definition students and engineers actually type into a form.

What you can do with it

Homework and lab checks

Compute vg ≈ Δω/Δk from two assigned or measured (k, ω) samples before grading wave-packet worksheets or comparing to waveguide data.

Photonics and fiber design notes

Turn nearby dispersion samples into group velocity for pulse delay, walk-off estimates, and group-index notes that expect SI-consistent speeds.

FAQ

What is the difference between group velocity and phase velocity?

Phase velocity is vp = ω/k for a single monochromatic wave. Group velocity is vg = dω/dk (here estimated as Δω/Δk) and is the envelope speed of a wave packet. They coincide only in non-dispersive media where ω is proportional to k.

What happens if k1 equals k2?

Division by zero Δk is undefined, so equal wave numbers are rejected and not billed.

Which units should I use?

Use any consistent pair. With ω in rad/s and k in rad/m, vg is in m/s. The engine does not convert units; it only evaluates the finite-difference quotient.

How do I estimate group velocity from two measured points?

Subtract the two angular frequencies, subtract the two wave numbers, and divide the first difference by the second. In the example, 2e14 rad/s over 1e6 rad/m gives 2e8 m/s.

How far apart should the two wave numbers be?

Close enough that the slope between them is nearly constant, yet far enough apart that rounding noise and measurement uncertainty do not dominate the quotient. That balance is standard practice in numerical differentiation.

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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

POSThttps://api.kit.forhosting.com/optics/group-velocity-finite-difference

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/group-velocity-finite-difference \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"omega1":1000000000000000,"k1":10000000,"omega2":1200000000000000,"k2":11000000}'

Example request

{
  "omega1": 1000000000000000,
  "k1": 10000000,
  "omega2": 1200000000000000,
  "k2": 11000000
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "optics.group_velocity_finite_difference",
  "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

Per request$0.002

Published price — no tokens, no invented credits. A failed task is never charged.

Errors

HTTPCodeMeaning
401unauthorizedMissing or invalid API key.
402insufficient_balanceYour balance doesn't cover the task price.
404unknown_typeThat task type doesn't exist.
429rate_limitedToo many requests. Use the webhook instead of polling.

Technical notes

What group velocity means and how the finite difference works

The response echoes the four inputs, reports delta_omega and delta_k, returns group_velocity and the short alias vg, and includes formula, equation, method tag, message, and ordered steps you can paste into a worksheet.

Input contract, validation, and reproducible response shape

Magnitudes above the documented absolute ceiling are rejected so intermediate products cannot overflow into non-finite results. Failed validation is not billed. POST a JSON body with four required finite numbers: omega1 (aliases w1, angular_frequency1, omega_1), k1 (aliases wave_number1, kappa1, k_1), omega2 (aliases w2, angular_frequency2, omega_2), and k2 (aliases wave_number2, kappa2, k_2). Each field may arrive as a JSON number or as a numeric string in ordinary or scientific notation; missing fields, empty strings, non-numeric tokens, and non-finite values fail with invalid_input naming the field. The method field is always finite_difference_group_velocity so automation can distinguish this estimate from a future closed-form derivative endpoint. Pair the response with your own plot of ω versus k if you need a visual slope; this endpoint returns only the scalar estimate and the bookkeeping fields needed for audit trails.

Where a two-point group velocity API earns its keep

Students meet group velocity when a wave-optics or solid-state chapter introduces packets, when a lab measures ω at two nearby k points on a waveguide, or when a contest problem asks for energy transport speed without calculus software. Acoustics and ultrasonics workers estimate packet speed from two frequency–wave-number samples on a measured dispersion branch. Teachers can pre-generate answer keys for homework that supplies two (k, ω) pairs and asks for the chord slope, without trusting opaque calculators that confuse phase and group velocity. Developers call the same contract from scripts to seed deterministic test vectors, to normalize user-entered dispersion tables before a visualization, or to satisfy programming challenges that demand vg rather than a hand-waved sketch. Photonics and fiber designers convert measured or simulated dispersion samples into vg before estimating pulse walk-off, delay, or group-index approximations n_g equals c over vg in SI units. Product and education tools use the endpoint when a learner types only the two sample points and the UI must show meters per second with a transparent formula. On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split—free interactive exploration and a priced, identical API—matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with wave-number and wavelength converters when you start from λ, with phase-velocity tools when you need ω over k for comparison, and with other optics endpoints when you are building a small lab of free, deterministic wave calculators focused on dispersion and packet motion.

Integration use cases

Scripts and CI fixtures

Pin deterministic vg from fixed ω and k pairs in automated tests without re-implementing finite-difference slope logic by hand.

Technical questions

How do I calculate group velocity from two (k, ω) points?

Send finite omega1, k1, omega2, and k2. The API returns vg = (ω2 − ω1)/(k2 − k1) as group_velocity (and alias vg), plus Δω, Δk, formula, equation, and steps.

What does the group velocity calculator cost?

The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.

Read the full KIT documentation →