WAV-30

Group velocity

Envelope speed.

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GoverningGroup velocity

Governing equation

v_g=d\omega/dk

$dk \in \left[0.1, 40\right]$

where

dw: dw (rad/s)
dk: dk (1/m)
vg: Group velocity (m/s)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

v_g=d\omega/dk

$dk \in \left[0.1, 40\right]$

$dk = 0.1$ $vg = 120$

Along the curve

②

Domain of the variable — blue

dk_{\min} = 0.1,\quad dk_{\max} = 40

$dk \in \left[0.1, 40\right]$

$dk = 0.1$ $dk = 0.1$

Along the curve

③

Response curve — green

v_g=d\omega/dk

$dk \in \left[0.1, 40\right]$

$dk = 0.1$ $vg = 120$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}vg}{\mathrm{d}dk}

$dk \in \left[0.1, 40\right]$

$dk = 0.351$ $vg^{\prime} = -199.4$

Along the curve

Lecture brief

Historical brief

d’Alembert’s wave equation, Snell, the thin-lens maker, Doppler and Bragg interference are the classical optics-and-sound toolkit. The lab is propagation, image and shift. This sheet (WAV-30 — Group velocity) is the form associated with Group velocity. Working symbols: $dw$ , $dk$ $\rightarrow$ $vg$ . Envelope speed. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

vg

from

dw

dk

in Waves & optics via

v_g=d\omega/dk

Envelope speed. Use it when a real waves & optics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

dw = 12.000\,\mathrm{rad/s}

dk = 4.000\,\mathrm{1/m}

, the governing relation

v_g=d\omega/dk

yields

vg = 3.000\,\mathrm{m/s}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

dw dw12 rad/sdk dk4 1/m

Outputs

Group velocity vg3.000 m/s

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Envelope speed. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube