Apparent Depth

Rays leaving a submerged object bend away from the normal as they cross into air, and the eye — which has no mechanism for knowing that a ray was bent — traces them back along straight lines to a point higher up than the object actually is. The result is a virtual image at depth $d/n$. For water at $n = 1.333$, that puts everything at three quarters of its true depth. The derivation is one line once you allow small angles: Snell's law becomes $n_1\theta_1 \approx n_2\theta_2$ near the normal, and a little trigonometry on the two triangles gives $d' = d/n$ directly. It is the same $n$ as in Snell's law and in $n = c/v$ — one number doing three jobs.

A pool with a true depth of 2.0 m looks about 1.5 m deep. A coin lying under 30 cm of water appears at 22.5 cm. The effect is not confined to water: a scratch on the far face of a 12 mm glass plate at $n = 1.5$ appears only 8 mm in, which is exactly why a high-magnification microscope objective carries a correction collar for coverslip thickness — get it wrong by a few tens of micrometres and the image degrades visibly.

The view from the other side is more interesting than the view from above. A fish looking up sees the entire 180° world above the surface squeezed into a cone whose half-angle is the critical angle, 48.6°, so the whole sky, shoreline and everything above it arrives compressed into a circular window about 97° wide directly overhead. Outside that circle the fish sees only the underwater scene, reflected back down by total internal reflection. Divers call it Snell's window and it is the same physics as this page, read in the opposite direction.

The formula assumes you are looking straight down. It is a small-angle result, and at oblique angles the apparent depth is less than $d/n$ — the bottom of a pool looks shallower still toward the far end, which is a substantial part of why people misjudge depth at the edge. That also means the folk rule about spear fishing is related to this page but is not this calculation: aiming below the visible fish is a matter of the sideways displacement at an oblique angle, and $d/n$ only handles the vertical case. Two more points. The image is virtual, so there is no light at that shallower depth and a camera focused underwater will not find anything there. And $n$ is the index of the medium the object is in, with the observer in air; reverse the arrangement — something in air viewed from underwater — and it appears farther away by a factor of $n$, not nearer.