Reflection of Waves and Formation of Stationary Waves

A travelling wave is reflected whenever it meets a boundary -- a change from one medium to another, or the end of a bounded medium such as a string or an air column. At a fixed (rigid) boundary, such as a string end clamped to a rigid wall, the reflected wave undergoes an abrupt phase reversal of $\pi$ (its displacement is inverted relative to the incident wave) -- physically, because the rigid support exerts whatever reaction force is needed to keep that one point permanently at zero displacement, and the only way superposition of incident and reflected waves can guarantee zero displacement there at all times is if the reflected wave is the incident wave's mirror image. At a free boundary, by contrast, the wave reflects without any phase reversal. When an incident wave and its own reflected wave -- travelling in opposite directions through the same medium, with the same amplitude and frequency -- overlap, the principle of superposition produces not a new travelling wave but a stationary (or standing) wave: $y(x,t) = 2A\sin(kx)\cos(\omega t)$ In this pattern, the spatial factor $2A\sin(kx)$ fixes an amplitude that varies from point to point along the medium but does not itself move; every particle at a given $x$ simply oscillates up and down (via the $\cos(\omega t)$ factor) with that fixed local amplitude, and the whole pattern of large-swing and no-swing regions stays fixed in space rather than travelling. Points where $\sin(kx)=0$ have zero amplitude at every instant and are called nodes -- they never move at all; points where $\sin(kx)=\pm1$ have the maximum possible amplitude $2A$ and are called antinodes. Successive nodes (and, separately, successive antinodes) are spaced exactly $\lambda/2$ apart, while a node and its nearest antinode are spaced $\lambda/4$ apart. Unlike a progressive wave, a stationary wave does not transport energy from one end of the medium to the other - …