Reflection of a Plane Wave by a Plane Surface

The figure depicts a horizontal reflecting surface labelled $MN$. Above this surface, two wavefronts are shown: the incident wavefront $AB$ and the reflected wavefront $CE$. A normal (a dashed line perpendicular to $MN$) is drawn at the point of incidence $C$. The angle between the incident wavefront $AB$ and the surface $MN$ is labelled $i$, and the angle between the reflected wavefront $CE$ and the surface is labelled $r$. Points $A$, $B$, $C$, and $E$ are marked: $B$ is on the incident wavefront, $C$ is where $B$ meets the surface, $A$ is another point on the incident wavefront, and $E$ is the point where the reflected wavefront touches the secondary wavelet from $A$.

The physical idea is Huygens' principle applied to reflection. As the incident plane wavefront $AB$ moves downward, point $B$ reaches the surface $MN$ at $C$ after time $t$, covering a distance $BC = vt$ (where $v$ is the wave speed). Meanwhile, point $A$ acts as a source of secondary wavelets; in the same time $t$, a wavelet of radius $AE = vt$ spreads from $A$ back into the medium. The tangent line from $C$ to this wavelet gives the reflected wavefront $CE$. The geometry forces the triangles $EAC$ and $BAC$ to be congruent, leading to the law of reflection.

The key formula derived from this figure is the law of reflection:

where:

• $i$ is the angle of incidence (the angle between the incident wavefront $AB$ and the reflecting surface $MN$),
• $r$ is the angle of reflection (the angle between the reflected wavefront $CE$ and the reflecting surface $MN$). …

What the Figure Shows

The figure has three panels, each showing a vertical plane wavefront (parallel straight lines) approaching an optical element from the left. The wavefronts are drawn as equally spaced lines representing the crests of the wave. The optical element is shown in cross-section, and the outgoing wavefronts are drawn on the right side of each element.

• Panel (a) – Thin prism: The incoming vertical wavefront hits a triangular glass prism. The lower part of the wavefront travels through more glass (the thicker base of the prism) than the upper part. Because light slows down in glass, the lower portion is delayed more. The emerging wavefront is tilted — it is still a plane wave but now slanted relative to the incoming one. The prism is labelled, and the tilt direction shows that the wave bends toward the base of the prism.

• Panel (b) – Convex lens: The incoming vertical plane wavefront meets a biconvex lens. The central part of the wavefront passes through the thickest part of the lens and is delayed the most. The edges pass through thinner glass and are delayed less. The emerging wavefront is curved — it has a depression at the centre, forming a spherical wave that converges to a point labelled F (the focus). The lens is labelled, and the focus is marked on the right side.

• Panel (c) – Concave mirror: The incoming vertical plane wavefront strikes a concave mirror (curved inward). Upon reflection, the wavefront becomes spherical and converges to a point labelled F (the focus). The mirror is labelled, and the focus is shown in front of the mirror.

No axes or numerical scales are present; the figure is schematic, using wavefront lines to illustrate the change in shape and direction.

Physical Idea

The figure demonstrates how wavefronts change shape when they pass through or reflect from optical elements. The key principle is that the speed of light is slower in glass than in air (or vacuum). When a plane wave enters a medium of different thickness, different parts of the wavefront travel different distances through the slower medium, causing a time delay that distorts the wavefront.

• In the prism, the delay varies linearly across the wavefront, producing a tilt — this is the wave explanation of refraction.
• In the convex lens, the delay is greatest at the centre, so the wavefront becomes concave (depressed at the centre), which makes it converge to a point — this is how a lens focuses light.
• In the concave mirror, reflection changes the direction of propagation, and the curved surface shapes the reflected wavefront into a converging spherical one.

The figure also illustrates a deeper principle: the total time taken by light to travel from a point on the object to the corresponding point on the image is the same along any ray. For example, in the convex lens, the central ray travels a shorter path in air but spends more time in the slower glass, while the edge rays travel longer in air but less in glass — the times balance, so all rays arrive at the focus simultaneously.

Key Formula

The textbook uses this figure to develop the law of reflection and the law of refraction (Snell's law) via Huygens' principle. The relevant formula for the time delay in a medium is:

where $v$ is the speed of light in glass. For a wavefront incident at an angle, the geometry leads to Snell's law: …