WAV-31

Phase velocity

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GoverningPhase velocity

Governing equation

v_p=\omega/k

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

where

w: w (rad/s)
k: k (1/m)
vp: Phase 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_p=\omega/k

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

$k = 0.1$ $vp = 120$

Along the curve

②

Domain of the variable — blue

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

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

$k = 0.1$ $k = 0.1$

Along the curve

③

Response curve — green

v_p=\omega/k

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

$k = 0.1$ $vp = 120$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}vp}{\mathrm{d}k}

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

$k = 0.351$ $vp^{\prime} = -199.4$

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-31 — Phase velocity) is the form associated with Phase velocity. Working symbols: $w$ , $k$ $\rightarrow$ $vp$ . Crest speed. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

vp

from

w

k

in Waves & optics via

v_p=\omega/k

Crest 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

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

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

, the governing relation

v_p=\omega/k

yields

vp = 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

w w12 rad/sk k4 1/m

Outputs

Phase velocity vp3.000 m/s

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Narration of this film

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

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

Reading speed

Watch on YouTube