Apparent Depth: Why Water Looks Shallower

Pour water into the mug and the coin hidden by the rim becomes visible. Move the eye and watch how rays refract at the water surface, where the eye “sees” the coin and why the apparent depth h′ = h/n is smaller than the real one. Dip a pencil into the water and it looks broken.

Experiments

Object

H liquid level cm

Liquid

n refractive index

Image and extended rays

Fan of rays from the coin

Test yourself: predict the result

Drag the eye to look from different angles. The image is built exactly: the extensions of two neighbouring rays entering the eye meet at one point. Looking from above gives the textbook formula h′ = h/n; looking obliquely lifts the bottom even more.

The experiment with a coin at the bottom of a bowl was described already in ancient treatises on optics. Put a coin into an opaque mug and step back until the rim hides it. Ask someone to pour water in slowly — the coin “rises” and appears, although neither it nor you have moved. Press “Pour water” and see at what level the ray from the coin first passes over the rim. Then move the eye: the more obliquely you look, the higher the bottom seems. That is why swimming in an unfamiliar pond is dangerous — it looks shallower — and why a spear-fisher aims below the fish he sees.

KEY FORMULAS

Law of refraction (Snell’s law)

$$\frac{\sin\alpha}{\sin\beta} = \frac{n_2}{n_1} \quad\Leftrightarrow\quad n_1\sin\alpha = n_2\sin\beta$$

α is the angle of incidence, β the angle of refraction (both measured from the normal to the boundary); n₁ is the refractive index of the medium the light comes from, n₂ of the medium it enters

▶ Detailed Explanation

The law follows from the fact that light travels at different speeds in different media. A wavefront hitting the boundary obliquely has one edge already in the second medium, slowed down, while the other edge still runs in the first — so the front turns (Huygens’ principle). In the same time the wave covers distances v₁t and v₂t, hence sin α / sin β = v₁/v₂ = n₂/n₁.

Example: a ray from air strikes glass (n = 1.5) at 40°. sin β = sin 40° / 1.5 = 0.643 / 1.5 = 0.429, β ≈ 25.4°. Conversely, from glass into air at 25.4° the ray leaves at 40° — light paths are reversible. The incident ray, the refracted ray and the normal lie in one plane.

Absolute refractive index

$$n = \frac{c}{v}, \qquad \lambda = \frac{\lambda_0}{n}$$

c = 3·10⁸ m/s is the speed of light in a vacuum, v the speed of light in the medium, λ₀ the wavelength in a vacuum, λ in the medium

▶ Detailed Explanation

The refractive index shows how many times slower light travels in a substance than in a vacuum: for air n ≈ 1.0003 (problems use 1), for water 1.33 (v ≈ 2.26·10⁸ m/s), for glass about 1.5 (v = 2·10⁸ m/s), for diamond 2.42 (v ≈ 1.24·10⁸ m/s). A medium with a larger n is called optically denser.

When light passes from one medium to another, its frequency ν does not change (as many crests leave the boundary as arrive), but the speed v = λν decreases — so the wavelength decreases too: λ = λ₀/n. Switch on the wavefronts in the simulator: in glass they are closer together and slower.

Apparent depth (view from above)

$$h' = \frac{h}{n}$$

h is the real depth, h′ the apparent depth, n the refractive index of the liquid relative to air; the formula holds when looking almost vertically

▶ Detailed Explanation

A ray from a point on the bottom leaves the water at a distance a from the vertical: a = h·tan β = h′·tan α. Hence h′ = h·tan β / tan α. For small angles the tangents are almost equal to the sines, and sin α / sin β = n, so h′ = h/n.

For water n = 1.33: a bottom 2 m deep seems to be 1.5 m deep, and a “knee-deep” river may really be waist-deep. The image lies on the same vertical as the object but n times closer to the surface. For glass (n = 1.5) a letter under a 3 cm thick block seems to be 2 cm deep.

Apparent depth for an oblique view

$$h' = \frac{h\cos^3\alpha}{n\cos^3\beta}, \qquad \sin\alpha = n\sin\beta$$

α is the angle between the ray entering the eye and the vertical; β is the angle of the same ray in the water. For α → 0 this gives h′ = h/n

▶ Detailed Explanation

The eye sees a point where the extensions of two neighbouring rays entering the pupil intersect. The ray leaves the surface at x = h·tan β, and its underwater extension is a straight line with slope tan α. Intersecting neighbouring lines (derivative with respect to β) gives the image depth h′ = (dx/dβ)/(d tan α/dβ) = h·cos³α/(n·cos³β).

Example: water, viewing angle α = 45°. sin β = 0.707/1.33 = 0.532, β ≈ 32.1°; h′/h = 0.354/(1.33·0.608) ≈ 0.44 — the bottom seems 2.3 times closer, while from above only 1.33 times. The image also shifts towards the observer. That is why the bottom of a pond seen into the distance seems to rise, and a fish looks higher and closer than it really is.

Why the bottom seems closer. Light from an underwater object refracts at the surface as it leaves the water and bends away from the normal: sin α = n·sin β, where n is the refractive index of the liquid relative to air. The eye knows nothing about refraction: it extends the incoming rays backwards in straight lines and sees the object where these extensions intersect. That point lies above the real object — the bottom is “lifted”.

The formula h′ = h/n. When you look almost from above, the angles are small and the tangents are almost equal to the sines. Then the law of refraction gives the apparent depth h′ = h/n. For water n = 1.33, so a bottom 2 m deep seems to be 1.5 m deep. For an oblique view the effect is stronger: the image in the plane of the drawing lies at depth h′ = h·cos³α / (n·cos³β), where α is the viewing angle from the vertical and β the angle of the ray in water. The simulator builds the image with this exact formula, and the graph shows how h′/h falls as the angle grows.

A coin in a mug. When the mug is empty, the ray from the coin to the eye goes in a straight line and hits the wall. Water changes the ray path: leaving the water, it tilts and passes over the rim — the coin becomes visible. Choose the “Coin in a mug” experiment, press “Pour water” and watch the status in the top-left corner.

The “broken” pencil and fishing. Every point of the underwater part of a pencil seems closer to the surface, so the pencil looks broken at the boundary between water and air. For the same reason a fish looks higher than it really is. Change the liquid: the larger n (ethanol, glycerol, carbon disulfide), the stronger the “lift”. Looking straight down, the coin stays on the same vertical, only n times closer.

Apparent Depth: Why Water Looks Shallower

KEY FORMULAS

Law of refraction (Snell’s law)

Absolute refractive index

Apparent depth (view from above)

Apparent depth for an oblique view

More on this topic

Other simulators

Problems on this topic