A standing wave forms when two waves of the same frequency and wavelength travel in opposite directions and interfere.
The result is a pattern that does not translate through space; instead, each point oscillates in time with an amplitude that depends on position.
A common form is
\[
y(x,t)=Y(x)\cos(\omega t),
\]
where \(Y(x)\) is the spatial “mode shape.” Points where \(Y(x)=0\) are nodes, meaning the displacement is always zero. Points where
\(|Y(x)|\) is maximal are antinodes.
The node/antinode locations depend on boundary conditions. In a string fixed at an end, the displacement must be zero there, so the end is a
node. In an open end of an air column (idealized), the displacement is maximal, so the end behaves like an antinode. These constraints determine which
wavelengths “fit” into the length \(L\).
Both ends fixed (string). The ends at \(x=0\) and \(x=L\) are nodes. The allowed wavelengths satisfy
\[
L=n\frac{\lambda}{2}\quad \Rightarrow\quad \lambda=\frac{2L}{n},\qquad n=1,2,3,\dots
\]
For these modes, nodes occur every \(\lambda/2\):
\[
x_m=m\frac{\lambda}{2}=m\frac{L}{n},\quad m=0,1,\dots,n,
\]
and antinodes lie halfway between nodes:
\[
x_m=\left(m+\frac{1}{2}\right)\frac{\lambda}{2}=\left(m+\frac{1}{2}\right)\frac{L}{n},\quad m=0,1,\dots,n-1.
\]
Example: \(L=1\) m, \(n=2\) gives \(\lambda=1\) m, nodes at \(0,0.5,1\) m and antinodes at \(0.25,0.75\) m.
Both ends open (air column). The ends act like antinodes. The wavelength condition is the same,
\[
\lambda=\frac{2L}{n},
\]
but the pattern is shifted: antinodes at \(x=mL/n\) and nodes at \(x=(m+1/2)L/n\).
One end fixed, one end open. This is a quarter-wave system. The fixed end is a node and the open end is an antinode, giving
\[
L=(2n-1)\frac{\lambda}{4}\quad \Rightarrow\quad \lambda=\frac{4L}{2n-1},\qquad n=1,2,3,\dots
\]
Only odd harmonics appear in this case. Node/antinode positions follow from quarter-wave spacing.
This calculator uses these standard 1D formulas to list node/antinode locations and draw the mode shape. More advanced systems include 2D membranes
and plates, where nodes become nodal lines and the patterns can be much more complex.
