Geometry decides the law
● VERBATIM — examined word-for-word3-D spherical (point source, "all directions"): I = P/4πr², I ∝ 1/r², A ∝ 1/r. Hemisphere ("all directions in front"): P/2πr². 2-D ripple (pond surface): energy over circumference → I ∝ 1/r, A ∝ 1/√r — DHS prints the warning: the 3-D law CANNOT be used. 1-D beam/plane (laser): I, A constant. Only I ∝ A² is universal.
spherical: I = P/4πr² · ripple: I ∝ 1/r · beam: I constWhat this actually means
The inverse-square law is not a law of waves; it is a law of spheres. A point source spreads its power over area 4πr², so I = P/4πr² and amplitude falls as 1/r.
Change the geometry and the law changes with it. 'All directions in front' means a hemisphere: P/2πr². Pond ripples spread around a circumference, so I ∝ 1/r and A ∝ 1/√r. A laser beam barely spreads at all: I stays constant.
Read the geometry cue words before writing any r-dependence. The only relation that survives every geometry is I ∝ A².
Using the 3-D inverse-square law for a 2-D surface ripple — ripples give I ∝ 1/r, A ∝ 1/√r.
Prove it — watch it be true
- In sphere mode, press r ×2: intensity drops to a quarter
- Switch to ripple mode, press r ×2: intensity only halves
- Check the log-log plot: slope −2 for the sphere, −1 for the ripple