Q.Explain how a stationary wave is formed by the superposition of an incident wave and its own reflected wave. Define the terms node and antinode, and state the spacing between two successive nodes in terms of the wavelength.

When a progressive wave meets a boundary, part or all of it is reflected back into the medium, travelling in the opposite direction to the incident wave but with the same amplitude and frequency (for a fully reflecting boundary). By the principle of superposition, the incident wave and its own reflected wave -- two waves of equal amplitude and frequency travelling in opposite directions through the same region -- combine, and the mathematics of this combination (adding $A\sin(kx-\omega t)$ and a suitably phase-adjusted $A\sin(kx+\omega t)$) produces $y(x,t)=2A\sin(kx)\cos(\omega t)$: a pattern whose spatial amplitude factor $2A\sin(kx)$ is fixed in position, even though each particle still oscillates in time via the $\cos(\omega t)$ factor. This fixed spatial pattern is the stationary (standing) wave.

A node is a point where $\sin(kx)=0$, so the amplitude of oscillation is zero there at every instant -- the particle at a node never moves at all. An antinode is a point where $\sin(kx)=\pm1$, the maximum possible value, so the particle there oscillates with the largest amplitude in the whole pattern, $2A$. …

Combine the incident and reflected waves via superposition to obtain the standing-wave form $y=2A\sin(kx)\cos(\omega t)$, then read off the node/antinode cond …

• Saying nodes and antinodes are spaced $\lambda$ apart instead of $\lambda/2$.
• Describing a node as a point where the wave 'stops travelling' rather than correctly as a point of perm …