Stationary waves

Stationary (standing) wave. Formed by superposition of two progressive waves of same:

• frequency,
• amplitude,
• speed, travelling in opposite directions.

Often: incident wave + reflected wave.

Properties.

• Pattern of nodes and antinodes is stationary.
• No net energy transfer.
• Distance between adjacent nodes = $\lambda/2$.
• Distance between adjacent antinodes = $\lambda/2$.
• Distance from node to nearest antinode = $\lambda/4$.

vs progressive wave.

• Progressive: transfers energy; all points have same amplitude.
• Stationary: no net energy transfer; amplitude varies along wave.

Cambridge tip. State the two requirements explicitly.

Both ends fixed. Nodes at each end.

Fundamental (1st harmonic): one antinode in middle. $\lambda = 2L$.

Higher harmonics: $\lambda_n = 2L/n$, $f_n = nv/(2L)$ for $n = 1, 2, 3, \ldots$.

Example. Guitar string 0.65 m, fundamental 220 Hz: $\lambda = 1.30$ m, $v = 286$ m/s.

Cambridge tip. Both string and pipe with two open ends have $f_n = nv/(2L)$.

Open end = antinode (free to vibrate). Closed end = node (cannot vibrate).

Pipe open at both ends. Antinodes at both ends. $f_n = nv/(2L)$, $n = 1, 2, 3, \ldots$. All harmonics.

Pipe closed at one end. Node at closed, antinode at open. Fundamental: $L = \lambda/4$ → $\lambda = 4L$.

$f_n = (2n-1)v/(4L)$, $n = 1, 2, 3, \ldots$. ONLY ODD harmonics ($f, 3f, 5f$, etc.).

Example. Closed pipe 0.34 m, $v = 340$ m/s: fundamental $\lambda = 1.36$ m → $f = 250$ Hz.

Cambridge tip. Closed pipe only has ODD harmonics — key distinction.