Open pipe fundamental: antinodes at both ends → 1 node at centre
Closed pipe fundamental: node at closed end, antinode at open end → 1 node total
Both have 1 node in fundamental mode → Ratio = 1:1
Answer: 1:1
Standing waves form by superposition of incident and reflected waves. Pressure and displacement nodes/antinodes: at an open end → displacement antinode, pressure node. At a closed end → displacement node, pressure antinode. Open pipe (length L): all harmonics: $f_n = nv/2L$ for $n = 1, 2, 3...$. nth harmonic: n antinodes at... wait: actually nth harmonic has (n+1) antinodes and n nodes (internal). Fundamental (n=1): 2 antinodes (ends) + 1 node (centre). Closed pipe: only odd harmonics: $f_n = nv/4L$ for $n = 1, 3, 5...$. Fundamental: 1 antinode (open end) + 1 node (closed end).
Speed of sound: $v = \sqrt{\gamma P/\rho} = \sqrt{\gamma RT/M}$. In air at 0°C: $v \approx 331$ m/s; at temperature $T$ K: $v \approx 331\sqrt{T/273}$. Speed increases with temperature, humidity. Frequency (pitch): $f$ = number of oscillations per second (Hz). Wavelength: $\lambda = v/f$. Intensity: $I \propto A^2 f^2$; Decibel: $\beta = 10\log(I/I_0)$ where $I_0 = 10^{-12}$ W/m².
Two waves of slightly different frequencies $f_1$ and $f_2$ superpose → beat frequency $f_{beat} = |f_1 - f_2|$. Maximum amplitude when waves in phase, minimum when out of phase. Used for tuning musical instruments: if beats heard → frequencies not equal; reduce beats until zero. Doppler effect: apparent frequency changes when source or observer moves. $f_{obs} = f_s\left(\frac{v \pm v_o}{v \mp v_s}\right)$ (+ for approach, − for recession).
Resonance: system vibrates at maximum amplitude when driving frequency matches natural frequency. In a pipe, resonance occurs at frequencies satisfying boundary conditions. Guitar string: $f_n = nv/2L = \frac{n}{2L}\sqrt{T/\mu}$ where $T$ = tension, $\mu$ = linear mass density. Higher tension, shorter string, lighter string → higher frequency (pitch). Harmonics in musical instruments create timbre (quality of sound). Overtones are frequencies above fundamental.