Nodes & Antinodes

To calculate the wavelength ($\lambda$) of a stationary wave, measure the distance between two adjacent nodes (or two adjacent antinodes) and multiply by two, as this distance represents exactly half a wavelength ($\frac{\lambda}{2}$).

The distance between a node and the immediately adjacent antinode is exactly one-quarter of a wavelength ($\frac{\lambda}{4}$), which is useful for analyzing systems with one open and one closed end.

When identifying the harmonic number ($n$) for a string fixed at both ends, count the number of antinodes (loops); the number of nodes will always be $n + 1$.

To determine the phase relationship between two points, identify if they lie within the same "loop" (between the same two nodes); if they do, they are in phase. If there is exactly one node between them, they are $180^\circ$ out of phase.

In a progressive wave, all points eventually reach the same maximum amplitude, whereas in a stationary wave, the amplitude is fixed for each specific point based on its distance from a node.

Energy in a stationary wave is stored within the oscillations between nodes, rather than being transferred through the medium as it is in a progressive wave.

Count Carefully: When asked for the number of nodes in a string fixed at both ends, remember to include the nodes at the very ends of the string, not just the ones in the middle.

Wavelength vs. Length: Always check if the given length $L$ represents a full wavelength or a fraction. For the fundamental frequency of a fixed string, $L = \frac{\lambda}{2}$, meaning $\lambda = 2L$.

Phase Logic: Remember that all particles between two adjacent nodes reach their maximum displacement at the same time, even if those displacements are different magnitudes. This means they are in phase.

Boundary Conditions: Always identify the boundaries first. A fixed end or a closed pipe end MUST be a node, while a free end or an open pipe end MUST be an antinode.

The Half-Wavelength Trap: A common mistake is assuming the distance between two nodes is a full wavelength. It is only $\frac{\lambda}{2}$. A full wavelength requires three consecutive nodes.

Displacement vs. Pressure: In longitudinal sound waves, a displacement node (where air doesn't move) is actually a pressure antinode (where pressure change is maximum). Ensure you know which property is being measured.

Motion at Nodes: Students often think nodes move slightly. By definition, in an ideal stationary wave, the displacement at a node is exactly zero at all times.