• Pixel-art sticker of Shallox, personifying Apparent Depth, with the caption “Deeper Than I Look.”

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Shallox

Apparent Depth Physics

\[d' = \frac{d}{n}\]
$4.99 USD
$4.99 USD
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Meet Shallox, a pebble on the bottom of a pool who is always deeper than he looks. Light bends as it leaves the water, so seen from above the apparent depth is the real depth divided by 1.33. A pool 2 m deep looks only about 1.5 m deep. Pixel-art kiss-cut sticker in 2, 3, 4 and 6 inches; equation, symbols and sources in the physics panel below.

Size
Surface

Flat $6 shipping worldwideMade to order, ships in 2–5 business days

Details

Size3″ × 3″

3″ × 3″ kiss-cut sticker.

Shipping

Flat $6 shipping on every order, worldwide, whatever's in it.

Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

Meet Shallox, a pebble on the bottom of a pool who is always deeper than he looks.

Light from the pebble travels up through the water and bends as it leaves the surface into the air. Because light travels more slowly in water, the ray bends away from the vertical as it escapes. Your eyes and brain trace the ray back in a straight line, so they place the pebble higher than it really is.

Looking from directly above, the apparent depth is d′ = d/n, where n is the refractive index of water, about 1.33. A pebble on the bottom of a pool 2 m deep appears to be only about 1.5 m down.

Look from a slant and the effect gets even stronger, which is why a straight stick poking into water seems to bend at the surface.

Anyone spearfishing or reaching for a dropped ring has to aim below where the object seems to be, and it is a good reason never to judge pool depth by eye before diving.

Equations

\[d' = \frac{d}{n}\]
SymbolMeaningUnit
\(d'\) apparent depth \(\mathrm{m}\)
\(d\) real depth \(\mathrm{m}\)
\(n\) refractive index of the water \(\mathrm{dimensionless}\)

Sources

  1. OpenStax University Physics Vol. 3 — Refraction
  2. Wikipedia — Refractive index

Field notes

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