Reflection (Wave Phenomena)

Change of direction of a wave when it bounces from a boundary. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Reflection occurs when a wave reaches a boundary and part of its disturbance returns to the original medium. It is common in light, sound, water waves, strings, and electromagnetic signals. The central idea is not only that the wave “bounces”, but that the boundary imposes an outgoing direction and may return only a fraction of the intensity.

This leaf focuses on two connected questions: the direction of the reflected wave and the amount of energy that returns. The relevant quantities are the angle of incidence $\theta_i$, the angle of reflection $\theta_r$, the incident intensity $I_i$, the reflected intensity $I_r$, and the reflectance $R$.

Reflection underlies mirrors, echoes, radar, sonar, optical fibers, acoustic insulation, and surface reading. It also prepares refraction, interference, and standing waves, because each reflection may change direction, phase, and intensity.

🟢 Essential level

Imagine a wave reaching a smooth wall. If the wall behaves regularly, the wave leaves with a clear symmetry about a line perpendicular to the wall, called the normal. That line is the correct reference for measuring angles.

The intuitive rule is simple: the wave leaves as tilted from the normal as it arrived. If it arrives almost head-on, it returns almost along itself. If it arrives at a grazing angle, it also leaves grazing on the other side of the normal.

[!TIP] Do not measure the angle from the surface. The law of reflection always uses the local normal.

Reflection also does not mean that all energy returns. Glass reflects part of the light and transmits another part; a wall returns part of the sound and absorbs part. Therefore reflected direction and reflected intensity are different questions.

A bathroom mirror and a window pane illustrate both extremes: the mirror is designed to reflect almost all the light that reaches it, while the pane lets most of the light through and reflects only a small fraction, enough to show a faint reflection of the room.

🔵 Formal level

Basic formulas

For specular reflection at a flat boundary smooth compared with the wavelength, the fundamental angular relation is:

Geometric law of reflection

This relation says that $\theta_r$ and $\theta_i$ are measured from the same normal. It does not compare angles from the surface or absolute directions in space. The normal acts as the local axis of symmetry between the incident ray and the reflected ray.

In practice, this angular equality must be applied only after the geometric convention is fixed. If the initial data are given from the surface, the angle from the normal must be reconstructed first; mixing references places the reflected ray in a physically different direction.

Real student questions

Why is the angle measured from the normal?

Because the normal represents the local orientation of the boundary. Measuring from the surface uses the complementary angle and breaks the correct symmetry.

Does every reflection conserve intensity?

No. Direction may obey the law even when $R$ is low and most energy is transmitted or absorbed.

Does a rough surface not reflect?

It does reflect, but it spreads the wave into many directions. This is diffuse reflection and is not described by one single $\theta_r$.

Does reflection always change phase?

Not always. It depends on the boundary and wave type. The angular law alone does not contain that information.

Final synthesis

Reflection combines a geometric rule and an energy reading. The angular law predicts direction through $\theta_i$ and $\theta_r$, while $R$ and $I_i$ determine the intensity that actually returns as $I_r$.