What happens when two waves arrive at the same point at the same time? The principle of superposition answers this: when two or more waves overlap, the resultant displacement at any point is the vector sum of the individual displacements. The waves pass through each other and emerge unchanged — they do not collide or scatter. This single principle underlies interference, beats and standing waves.
Interference is the result of superposing two waves of the same frequency travelling in the same direction:
- Constructive interference occurs when the waves arrive in phase (crest meets crest). The amplitudes add, giving a louder sound or brighter light. This needs a path difference of a whole number of wavelengths: $\Delta x=n\lambda$.
- Destructive interference occurs when the waves arrive out of phase by half a cycle (crest meets trough). The amplitudes cancel, giving silence or darkness. This needs a path difference of an odd number of half-wavelengths: $\Delta x=(2n+1)\frac{\lambda}{2}$.
Reflection of waves. When a wave hits a boundary it is reflected. The nature of the reflection depends on the boundary:
- At a rigid (fixed) boundary, the reflected wave suffers a phase change of $\pi$ (a crest returns as a trough). A node always forms there.
- At a free (open) boundary, the reflected wave has no phase change. An antinode forms there.
Standing (stationary) waves form when two identical waves travel in opposite directions and superpose — typically an incident wave and its reflection. The result is a pattern that appears to stand still: $y=2A\sin(kx)\cos(\omega t)$. Energy is not transported along the medium; it stays trapped. The pattern has:
- Nodes: points of zero displacement (always at rest). Spacing between consecutive nodes is $\frac{\lambda}{2}$.
- Antinodes: points of maximum displacement (vibrating most). They lie midway between nodes.
Standing waves on a string fixed at both ends (length $L$): both ends must be nodes, so only certain wavelengths fit. The allowed frequencies are $f_n=\frac{n}{2L}\sqrt{\frac{T}{\mu}}$ for $n=1,2,3,\dots$ The lowest is the fundamental (first harmonic), $f_1=\frac{1}{2L}\sqrt{\frac{T}{\mu}}$, and the higher ones ($2f_1, 3f_1, \dots$) are harmonics. An overtone is any frequency above the fundamental; the first overtone is the second harmonic.
Air columns. Wind instruments use standing waves of sound:
- An open pipe (open at both ends) has antinodes at both ends. It produces all harmonics: $f_n=\frac{nv}{2L}$ ($n=1,2,3,\dots$).
- A closed pipe (one end closed) has a node at the closed end and an antinode at the open end. It produces only odd harmonics: $f_n=\frac{nv}{4L}$ ($n=1,3,5,\dots$). Its fundamental is $\frac{v}{4L}$ — half the open pipe's, which is why a closed pipe of the same length sounds an octave lower.