3.4 Reflection at a boundary

A right-going pulse on a semi-infinite string approaches the end $x = 0$, which is clamped — held fixed. What happens?

The clamping enforces $y(0, t) = 0$ for all $t$. The d’Alembert decomposition $y(x, t) = F(x - ct) + G(x + ct)$ must respect this constraint, so for all $t$,

equivalently, $G(\eta) = -F(-\eta)$ for all $\eta$. The left-going wave is the negative of the right-going wave, spatially mirrored. Whatever shape $F$ has, $G$ has the upside-down, left-right-flipped version. As $F$ propagates into $x = 0$, it generates a $G$ that propagates back outward.

A clamped end inverts

For a pulse approaching a clamped (fixed) end, the reflected pulse is inverted. A bump on the way in becomes a dip on the way out. The mathematical reason: $y(0, t) = 0$ forces the incident and reflected fields to cancel exactly at the boundary, which requires inversion.

A free end does not invert

For a free end — one that is allowed to move but is mass-less and has no spring (so its slope must vanish: $\partial_x y(0, t) = 0$) — the boundary condition forces $F'(-ct) + G'(ct) = 0$, i.e. $G'(\eta) = -F'(-\eta)$. Integrating, $G(\eta) = F(-\eta) + \text{const}$. The reflected pulse is flipped left–right but not inverted. A bump on the way in stays a bump on the way out.

In between: a general impedance

A real boundary often lies between “clamped” and “free.” A massive bead at the end, or a string joining to a thicker string, or a string anchored to a spring — each provides partial reflection. The boundary then has a reflection coefficient $R$ between $-1$ (clamped: full inversion) and $+1$ (free: no inversion), with $|R| < 1$ for any boundary that transmits or absorbs some of the wave’s energy. We will derive $R$ in terms of acoustic impedance in chapter 7.

The interactive

The figure above (originally drawn for the cochlea but recast here as a pure travelling-wave demo) shows a pulse moving along a 1-D medium with a partially reflecting boundary on the right. Vary the wave speed and the boundary impedance and watch the reflection coefficient change.

Two ends — the standing wave is built

If both ends of a string are clamped, every disturbance reflects off both walls, inverts at each, and propagates back. After many bounces the right-going and left-going components interfere. For special frequencies — where each round-trip lands the wave back in phase with itself — the bouncing reinforces, and a standing wave emerges. That is the topic of the next lesson.