Wave Reflection — Phase Inversion, Fixed Ends and the Law of Reflection

A pulse meets a wall

Send a single pulse down a stretched rope toward a wall and watch what comes back. The pulse does not vanish into the wall — it returns toward you, traveling through the same rope it came down. That return journey is reflection: a wave bouncing off a boundary back into its original medium.

What makes reflection interesting is not that the pulse comes back, but how it comes back. The boundary's nature — rigid or free, head-on or angled — writes itself into the reflected wave.

Fixed end: the inverted reflection

Tie the rope firmly to a wall and send an upward pulse toward it. The reflected pulse comes back downward — inverted, as if flipped through the rope's resting line.

Why? The tied end cannot move, so the arriving pulse yanks upward on the wall, and the wall yanks back downward on the rope (Newton's third law). That downward reaction launches the return pulse upside down. In wave language, a fixed end imposes a phase shift of half a cycle ($\pi$ radians): crest becomes trough and trough becomes crest.

Free end: the upright reflection

Now thread the rope's end through a smooth ring that slides freely on a vertical pole — a free end. Send the same upward pulse. This time it returns upward, upright and unflipped.

The free end offers no resistance: the arriving pulse lifts the ring to double height momentarily, and the ring's fall launches the reflected pulse in the original orientation. There is no phase shift — crest stays crest. The same rule governs a sound wave hitting open air or a water wave reaching a gentle beach: softer boundaries reflect upright.

Oblique incidence: the law of reflection

Waves rarely strike boundaries head-on. For a ray hitting a smooth surface at an angle, reflection obeys one clean rule:

$$\theta_i = \theta_r$$

The angle of incidence $\theta_i$ equals the angle of reflection $\theta_r$ — and both are measured from the normal, the line perpendicular to the surface at the impact point, with incident and reflected rays on opposite sides of it.

This law holds for light on mirrors, sound on walls, and water waves on harbor barriers alike. It is why you can aim a billiard-style bank shot with a laser pointer — and why concert halls angle their wall panels to throw sound exactly where the audience sits.

Partial reflection at soft boundaries

Real boundaries are rarely perfect walls. When a wave meets the boundary between two media it can enter — like sound passing from air into water — part of the wave reflects and part transmits through. This is partial reflection.

How much reflects depends on the contrast between the media, called the impedance mismatch. Shouting across a canyon gives a strong echo because air and rock are wildly different; a doctor's ultrasound gel exists to reduce the mismatch between the probe and your skin, so more sound enters the body instead of reflecting off it.

Worked mini-example

A pulse travels at $10\ \mathrm{m/s}$ along a rope toward a fixed wall $2\ \mathrm{m}$ away.

• Time to reach the wall: $t = d/v = 2/10 = 0.20\ \mathrm{s}$.
• Time for the round trip back to the starting point: $0.40\ \mathrm{s}$.
• The returning pulse is inverted (fixed end) and slightly smaller (some energy lost as heat and sound at the wall).

If the end were free instead, the pulse would return in the same $0.40\ \mathrm{s}$ but upright. Same timing, opposite phase — the boundary type is written into the wave.

Everyday reflections

Reflection is a working tool, not just a textbook demo. Echo sounding maps the seafloor by timing sound pulses reflected off the bottom. Medical ultrasound builds images from reflections at tissue boundaries. Radar and lidar locate aircraft and map terrain with reflected radio waves and laser pulses.

Even architecture leans on it: a whispering gallery is a room shaped so that sound reflects along its walls to a distant listener, and anechoic chambers do the opposite — lining walls with wedges that swallow reflections so engineers can test speakers in pure, reflection-free sound.

Variables and units

Common mistakes

Real-world applications

• Sonar and medical ultrasound send pulses out and read their reflections to map what they cannot see.

• Radar guns bounce radio waves off cars; the reflected signal reveals the vehicle's speed.

• Concert-hall designers place reflective and absorptive surfaces so reflections enrich the music instead of muddying it.