C.4 Standing waves and resonance

Standing Wave: A standing wave is a wave pattern formed by the superposition of two identical waves travelling in opposite directions through the same medium. The resulting pattern appears stationary , hence the name.

When a wave travels along a string and reaches a fixed end, it reflects back. If the reflected wave has the same frequency, wavelength, and amplitude as the original, the two waves interfere continuously. This interference follows the principle of superposition: the resultant displacement at any point equals the sum of the displacements from each individual wave.

The result is a pattern that does not move through the medium , instead, certain points always stay still while others always vibrate with maximum displacement.

Node: A node is a point on a standing wave where the displacement is always zero. Nodes arise from complete destructive interference between the two component waves.

Antinode: An antinode is a point on a standing wave where the displacement oscillates between its maximum positive and maximum negative values. Antinodes arise from complete constructive interference, producing an amplitude equal to twice that of each individual wave.

Key spatial relationships:

• Consecutive nodes are separated by $\frac{\lambda }{2}$.
• Consecutive antinodes are separated by $\frac{\lambda }{2}$.
• The distance from a node to the nearest antinode is $\frac{\lambda }{4}$.

Boundary conditions determine whether a node or antinode forms at the ends of the medium:

• A fixed end is always a node (the medium cannot move there).
• A free end is always an antinode (the medium is free to oscillate maximally).