Standing Waves and Resonance

Two waves with the same frequency, amplitude, and wavelength travelling in opposite directions create a standing wave.

Node: Point of zero displacement (destructive). Separation between nodes = $\lambda/2$. Antinode: Point of maximum displacement (constructive). Midway between nodes.

$f_n = \frac{nv}{2L}$ ($n = 1, 2, 3, ...$)

Fundamental ($n=1$): $L = \lambda/2$, $f_1 = v/(2L)$

$v = \sqrt{T/\mu}$ where $T$ = tension, $\mu$ = mass per unit length.

Open both ends: $f_n = nv/(2L)$ — all harmonics. Antinodes at both ends.

Closed one end: $f_n = nv/(4L)$ ($n = 1, 3, 5, ...$) — odd harmonics only. Node at closed end, antinode at open end.

Resonance occurs when the driving frequency matches a natural frequency of the system → maximum amplitude energy transfer.

• Standing wave: two opposite waves → nodes (zero) + antinodes (max)
• String/open pipe: $f_n = nv/(2L)$, all harmonics
• Closed pipe: $f_n = nv/(4L)$, odd harmonics only
• Resonance: driving freq = natural freq → max amplitude