= Apparent depth
{title2=$d_{\mathrm{par}}=(n_{\mathrm{obs}}/n_{\mathrm{obj}})d$}
A submerged object looks shallower because its emerging <light rays> bend away from the surface <normal vector>. Their backward extensions give an approximate <virtual image> nearer the surface. For an object at perpendicular depth $d$ in a medium of <refractive index> $n_{\mathrm{obj}}$, viewed through a flat boundary from a medium of <refractive index> $n_{\mathrm{obs}}$, the near-normal <paraxial approximation> gives $d_{\mathrm{par}}=(n_{\mathrm{obs}}/n_{\mathrm{obj}})d$. Thus the model <water> depth $12\,\mathrm{cm}$ appears approximately $9\,\mathrm{cm}$ from air. This is a bundle-of-<light rays> statement, not a deflection of the axial <light ray>.<University Physics Volume 3 by Samuel J. Ling and others>[§2.3, Refraction at a Plane Interface—Apparent Depth, Figures 2.13–2.14 and equation 2.10: near-normal small-angle assumption]{ref}
For general transmitted <light rays>, specify what “depth” means. Put the object at $(0,-d)$ below a horizontal boundary $z=0$. A <light ray> reaches the boundary at $(x,0)$, with object-side <angle> $\theta_{\mathrm{obj}}$ and observer-side <angle> $\theta_{\mathrm{obs}}$, both measured from the upward <normal vector>. For $x>0$, its backward extension meets the object's normal line $x=0$ at $z=-d_{\mathrm{axis}}$, where
$$
x=d\tan\theta_{\mathrm{obj}},\qquad
n_{\mathrm{obj}}\sin\theta_{\mathrm{obj}}=n_{\mathrm{obs}}\sin\theta_{\mathrm{obs}},\qquad
d_{\mathrm{axis}}=\frac{x}{\tan\theta_{\mathrm{obs}}}
=d\frac{n_{\mathrm{obs}}\cos\theta_{\mathrm{obs}}}{n_{\mathrm{obj}}\cos\theta_{\mathrm{obj}}}.
$$
This axis-intercept depth is exact for that one transmitted <light ray> in the stated model. It tends to $d_{\mathrm{par}}$ as the <angles> tend to zero; at exactly zero the quotient is interpreted as a <limit>. These elementary <tangent>, <sine> and <cosine> relations show precisely where the <small-angle approximation> enters.
The original calculations below use $d=12\,\mathrm{cm}$, $n_{\mathrm{obj}}=4/3$ and $n_{\mathrm{obs}}=1$:
|| Outgoing air <angle> $\theta_{\mathrm{obs}}$
|| <Water> <angle> $\theta_{\mathrm{obj}}$
|| Boundary position $x$ in cm
|| Axis-intercept depth $d_{\mathrm{axis}}$ in cm
| $5^\circ$
| $3.7479^\circ$
| $0.7861$
| $8.9850$
| $30^\circ$
| $22.0243^\circ$
| $4.8542$
| $8.4078$
| $60^\circ$
| $40.5054^\circ$
| $10.2509$
| $5.9184$
\Image[/refraction-apparent-depth-angle-dependence.png]
{title=Exact <light ray> extensions and the near-normal apparent-depth approximation}
{description=Original model of an object $12\,\mathrm{cm}$ below a planar <water>–air boundary, with <refractive indices> $4/3$ and $1$. The left panel has equal coordinate scales; solid lines are forward <light rays> and dashed lines are backward extensions to the object's normal line. The two outgoing <light rays> are different viewing directions, not a bundle entering one eye. The right panel compares their exact axis-intercept depths with the near-normal value $9\,\mathrm{cm}$. Marked points are calculated predictions, not observations.}
At appreciable obliquity these depths differ, so the full outgoing bundle has no single exact <virtual image> on the object's normal line. An oblique eye samples a narrow bundle; its local apparent position also depends on that bundle and cannot be inferred by declaring $d_{\mathrm{axis}}$ to be a universal image depth. A plane refracting boundary gives approximate <image formation> near <normal incidence>, with departures at larger <angles>. The construction assumes a flat surface, uniform transparent media with <isotropy>, and a propagating transmitted <light ray>; it does not extend to <total internal reflection>.
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