Organ Pipe Resonance: Choosing the Right Harmonic Model
A student investigates resonance in a pipe using a tuning fork of frequency 512 Hz. With one end closed, the first strong resonance occurs when the air column is 16.0 cm long. The room temperature is 22 degrees C. The student also finds a second resonance at 48 cm. A classmate using an open pipe of the same length gets a different fundamental. The student is asked to determine the speed of sound and decide whether the pipe behaves as open-open or closed-open.
1. Use the resonance positions to determine the speed of sound and confirm the boundary condition (closed at one end vs. open at both ends).
Read the worked answer
For a pipe closed at one end, resonances occur at L = (2n-1) lambda/4, so successive resonance lengths differ by lambda/2. Here 48 - 16 = 32 cm = lambda/2, so lambda = 64 cm = 0.64 m. Then v = f lambda = 512 x 0.64 = 328 m/s.
Check the boundary condition: the first resonance at L = lambda/4 = 16 cm matches exactly, confirming a closed-open pipe (a pressure node at the open end, antinode at the closed end). If it were open-open, the first resonance would be at L = lambda/2 and the spacing would also be lambda/2; but then the first resonance would be 32 cm, not 16 cm. The 16 cm first resonance is the discriminator.
Temperature check: v = 331 + 0.6T = 331 + 13.2 = 344 m/s at 22 C. The measured 328 m/s is ~5% low, consistent with an end correction (effective length slightly longer than physical length) and the finite fork frequency tolerance.
2. The classmate's open pipe of the same 16 cm length is said to have a different fundamental. Calculate both fundamentals and explain the factor-of-two relationship physically.
Read the worked answer
Closed-open fundamental: f1 = v/(4L) = 328/(4 x 0.16) = 512 Hz (matches the fork, as designed). Open-open fundamental: f1 = v/(2L) = 328/(2 x 0.16) = 1025 Hz, about twice.
Physically, the closed end forces a displacement node there, so the shortest standing wave fits a quarter wavelength in the pipe; the open-open pipe fits a half wavelength, so for the same length its fundamental is an octave higher. Equivalently, the closed pipe supports only odd harmonics (f1, 3f1, 5f1...), while the open pipe supports all harmonics. This is why the same physical length gives different pitch and why the closed pipe's second resonance is at 3 x 16 = 48 cm (third harmonic), not at 32 cm.