WAV-28

Spherical intensity

Inverse-square sound/light.

Reading speed

GoverningSpherical intensity

Governing equation

I=P/(4\pi r^2)

$r \in \left[0.2, 40\right]$

where

P: P (W)
r: r (m)
I: Spherical intensity (W/m²)

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

I=P/(4\pi r^2)

$r \in \left[0.2, 40\right]$

$r = 0.2$ $I = 19.89$

Along the curve

②

Domain of the variable — blue

r_{\min} = 0.2,\quad r_{\max} = 40

$r \in \left[0.2, 40\right]$

$r = 0.2$ $r = 0.2$

Along the curve

③

Response curve — green

I=P/(4\pi r^2)

$r \in \left[0.2, 40\right]$

$r = 0.2$ $I = 19.89$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}I}{\mathrm{d}r}

$r \in \left[0.2, 40\right]$

$r = 0.45$ $I^{\prime} = -36.5$

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-28 — Spherical intensity) is the form associated with Spherical intensity. Working symbols: $P$ , $r$ $\rightarrow$ $I$ . Inverse-square sound/light. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

I

from

P

r

in Waves & optics via

I=P/(4\pi r^2)

Inverse-square sound/light. 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

P = 10.000\,\mathrm{W}

r = 4.000\,\mathrm{m}

, the governing relation

I=P/(4\pi r^2)

yields

I = 0.050\,\mathrm{W/m^{2}}

. 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

P P10 Wr r4 m

Outputs

Spherical intensity I0.050 W/m²

Reading speed

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Narration of this film

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

Inverse-square sound/light. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube