Reflection at Boundaries
When a wave travelling along a medium reaches the end of that medium — or the point where one medium meets another — it does not simply vanish. Instead, part or all of the wave bounces back. This phenomenon is called reflection at a boundary. It occurs with all types of waves: waves on strings, sound waves in pipes, light waves at mirrors, and water waves hitting a wall. Understanding reflection is fundamental to explaining standing waves, musical instrument acoustics, sonar, and optical instruments. There are two fundamentally different types of boundary. A fixed end (also called a hard boundary or rigid boundary) is one where the medium is not free to move — imagine a string tied tightly to a wall. A free end (also called a soft boundary) is one where the medium is free to move — imagine a string attached to a light ring that can slide freely along a frictionless pole. The type of boundary determines the phase of the reflected wave relative to the incoming wave, which has profound physical consequences. The key rule is simple: reflection at a fixed end produces a phase change of 180° (π radians), inverting the pulse. Reflection at a free end produces no phase change, so the pulse reflects without inversion. These two rules govern everything from how standing waves form on guitar strings to how sound resonates inside organ pipes.
- What happens at a fixed end
- What happens at a free end
- Amplitude and the law of reflection
- Superposition and standing waves
What happens at a fixed end
At a fixed end, the boundary point is constrained and cannot move. By Newton's third law, when the arriving wave exerts an upward force on the wall, the wall exerts an equal and opposite downward force on the string. This reaction launches a reflected wave that is inverted — its crest becomes a trough and vice versa. In terms of phase, the reflected wave is shifted by exactly 180° (or π radians) relative to the incident wave. Think of it like bouncing a ball off a hard floor: the ball compresses on impact and rebounds in the opposite direction.
What happens at a free end
At a free end, the boundary point is unconstrained and free to move. When a crest arrives, the end of the medium can overshoot beyond its equilibrium position because there is nothing to restrain it. The reflected wave therefore has the same orientation as the incident wave — a crest reflects as a crest, a trough as a trough. There is no phase inversion (phase change = 0). An analogy: imagine shouting into an open tube; the sound reflects back without being 'flipped' in pressure, which is why open-ended pipes have pressure antinodes at the open end.
Amplitude and the law of reflection
For a perfectly rigid fixed end or a perfectly free end on a string, the amplitude of the reflected wave equals the amplitude of the incident wave — all energy is reflected and none is transmitted (assuming the medium does not absorb energy). In practice, at a partial boundary (where two different media meet), some energy is transmitted and some reflected. The fraction reflected depends on the impedance mismatch between the two media. The greater the mismatch, the more energy is reflected.
Superposition and standing waves
The reflected wave travels back through the same medium as the incident wave. The two waves occupy the same space simultaneously, and by the principle of superposition, the net displacement at any point is the sum of both displacements. When the frequency and boundary conditions are just right, the incident and reflected waves interfere to produce a standing wave — a pattern of fixed nodes (zero displacement) and antinodes (maximum displacement). For a string fixed at both ends, nodes must exist at both boundaries; for a pipe open at one end, a node occurs at the closed end and an antinode at the open end.
The Key Equation
- \phi_{\text{reflected}} — Phase of reflected wave (radians): The phase angle of the wave after it has bounced back from the boundary.
- \phi_{\text{incident}} — Phase of incident wave (radians): The phase angle of the wave approaching the boundary.
- \pi — Phase shift at fixed end (radians): A phase shift of π radians (180°) corresponds to a complete inversion of the wave — crests become troughs.
Worked Example
Problem. A transverse pulse travels along a string towards a fixed wall. The pulse has an amplitude of 3.0 cm and a width of 20 cm. Describe the reflected pulse fully, and state what the net displacement of the string is at the exact moment the pulse is centred on the wall.
Answer. The reflected pulse has amplitude 3.0 cm but is inverted (trough). When centred on the wall, superposition gives zero net displacement at that point, which is consistent with the fixed-end constraint.
Origins · 1 minMarin Mersenne and how this was discovered
Mersenne was a 17th-century mathematician, theologian, and natural philosopher who conducted the first quantitative studies of vibrating strings and sound waves. He formulated Mersenne's laws, which relate the frequency of a vibrating string to its length, tension, and linear density. His work laid the foundation for understanding standing waves — the direct consequence of wave reflection at boundaries — and he correctly described overtone series in strings and pipes. Mersenne was friends with René Descartes and Galileo Galilei, and acted as an informal 'scientific postmaster' of Europe, relaying letters and ideas between the great thinkers of his era — effectively running the 17th-century equivalent of a scientific email list.
Related Concepts
- Standing Waves — Standing waves are formed directly by the superposition of an incident wave and its reflection from a boundary; reflection rules determine where nodes and antinodes appear.
- Superposition of Waves — When a reflected wave travels back through the incident medium, the two waves add by superposition — the fundamental principle that explains the patterns seen at boundaries.
- Resonance in Strings and Pipes — Resonance occurs at specific frequencies determined by boundary conditions — fixed or free ends — which dictate where nodes and antinodes must form via reflection.
- Transmission at Boundaries — At real boundaries, energy is both reflected and transmitted; understanding reflection is incomplete without also considering how much wave energy passes into the new medium.