KAHIBARO
Table of Contents
What stationary patterns look like
When two waves of the same frequency and amplitude travel in opposite directions through the same medium, they can form a standing wave. In a standing wave, the pattern does not travel along the medium. Instead, some points always remain at rest, while other points move back and forth with the largest possible motion. These special points are called nodes and antinodes.
A node is a point where the displacement is always zero. No matter when you look, that point does not move. An antinode is a point where the displacement reaches the greatest amplitude in the standing wave. These points oscillate the most.
If a standing wave on a string is written as
$$
y(x,t) = 2A \sin(kx)\cos(\omega t),
$$
then the motion depends on both position $x$ and time $t$. The factor $\sin(kx)$ tells us how large the oscillation is at each location. The factor $\cos(\omega t)$ tells us how it changes with time.
Locating nodes
Nodes occur wherever the position factor is zero:
$$
\sin(kx) = 0.
$$
This happens when
$$
kx = n\pi,
$$
so
$$
x = \frac{n\pi}{k}.
$$
Using $k = \frac{2\pi}{\lambda}$, this becomes
$$
x = n\frac{\lambda}{2},
$$
where $n = 0, 1, 2, 3, \dots$
This means that adjacent nodes are separated by half a wavelength.
For a standing wave, nodes occur at positions
$$
x = n\frac{\lambda}{2}
$$
and the distance between neighboring nodes is
$$
\frac{\lambda}{2}.
$$
Locating antinodes
Antinodes occur wherever the magnitude of the position factor is greatest:
$$
|\sin(kx)| = 1.
$$
This happens when
$$
kx = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \dots
$$
so
$$
x = \left(n + \frac{1}{2}\right)\frac{\lambda}{2}
$$
or equivalently
$$
x = (2n+1)\frac{\lambda}{4},
$$
where $n = 0, 1, 2, 3, \dots$
Thus adjacent antinodes are also separated by $\frac{\lambda}{2}$, and each antinode lies halfway between two nodes.
Antinodes occur at positions
$$
x = (2n+1)\frac{\lambda}{4}
$$
and each antinode is midway between two neighboring nodes.
Physical meaning
Nodes are places of permanent destructive interference. At those points, the two opposite waves always cancel exactly. Antinodes are places of permanent constructive interference. At those points, the two waves always combine to give the largest oscillation.
This does not mean that the string or medium is frozen. The standing wave still changes with time. The important idea is that the pattern of nodes and antinodes stays fixed in space.
How amplitude varies with position
In a traveling wave, every point usually has the same amplitude. In a standing wave, the amplitude depends on position. From
$$
y(x,t) = 2A \sin(kx)\cos(\omega t),
$$
the amplitude at position $x$ is
$$
A_{\text{standing}}(x) = 2A |\sin(kx)|.
$$
So the amplitude is zero at nodes and maximum, equal to $2A$, at antinodes.
In a standing wave formed by two waves of amplitude $A$, the maximum oscillation at an antinode is
$$
2A.
$$
The amplitude is not the same everywhere.
Spacing summary
The geometry of nodes and antinodes is very regular.
| Quantity | Value |
|---|---|
| Node to next node | $\frac{\lambda}{2}$ |
| Antinode to next antinode | $\frac{\lambda}{2}$ |
| Node to nearest antinode | $\frac{\lambda}{4}$ |
These distances are extremely useful when analyzing strings, air columns, and other systems that support standing waves.
Visual picture
A simple standing wave pattern on a string can be sketched as alternating nodes and antinodes along the length.
In this drawing, the points marked N stay fixed, while the points marked A move up and down with maximum amplitude.
Time behavior of nodes and antinodes
At a node, displacement is always zero:
$$
y_{\text{node}}(t) = 0.
$$
At an antinode, the displacement changes sinusoidally with the largest amplitude:
$$
y_{\text{antinode}}(t) = \pm 2A \cos(\omega t).
$$
Different antinodes can move in opposite directions at the same instant. One antinode may be at maximum upward displacement while the next is at maximum downward displacement. Even so, all antinodes oscillate with the same frequency.
Why this matters
Nodes and antinodes tell us where energy appears as motion and where the medium remains still. In real physical systems, this helps identify allowed vibration patterns. On strings, fixed ends are nodes. In air columns, depending on the boundary, the ends may be nodes or antinodes of displacement. The exact application belongs to separate topics, but the central idea remains the same: standing waves are recognized by their fixed nodes and antinodes.
A standing wave is identified by fixed positions of zero motion, called nodes, and fixed positions of maximum motion, called antinodes.
Nodes and antinodes do not travel with the wave pattern.
A quick example
Suppose the wavelength in a standing wave is $\lambda = 0.80\ \text{m}$.
Then the spacing between adjacent nodes is
$$
\frac{\lambda}{2} = 0.40\ \text{m},
$$
and the distance from a node to the nearest antinode is
$$
\frac{\lambda}{4} = 0.20\ \text{m}.
$$
So if one node is at $x = 0$, the next few important positions are:
| Position | Type |
|---|---|
| $0.00\ \text{m}$ | Node |
| $0.20\ \text{m}$ | Antinode |
| $0.40\ \text{m}$ | Node |
| $0.60\ \text{m}$ | Antinode |
| $0.80\ \text{m}$ | Node |
This repeating pattern is one of the clearest signatures of a standing wave.