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Stationary Sound Waves in Organ Pipes (Open & Closed Pipes) - Unit #12 Acoustics

Video Lecture: Stationary Sound Waves in Organ Pipes

Watch the complete physics lecture explaining standing sound waves, open versus closed organ pipes, boundary conditions, and harmonic frequency derivations:

Stationary Sound Waves in Organ Pipes

Musical instruments like flutes, clarinets, and pipe organs rely on air columns inside pipes to produce sound waves. When sound waves reflect from the boundaries of a pipe, they interfere to form stationary (standing) sound waves.

1. Open Organ Pipes (Both Ends Open)

When both ends of a pipe are open, air molecules are free to vibrate, forming displacement anti-nodes at both ends. For a pipe of length $L$:

• Fundamental Mode (First Harmonic): $L = \frac{\lambda_1}{2} \implies f_1 = \frac{v}{2L}$
• Second Harmonic (First Overtone): $L = \lambda_2 \implies f_2 = \frac{v}{L} = 2f_1$
• Third Harmonic (Second Overtone): $L = \frac{3\lambda_3}{2} \implies f_3 = \frac{3v}{2L} = 3f_1$
• General Formula ($n$-th Harmonic): $f_n = n f_1 = \frac{n v}{2L} \quad (\text{where } n = 1, 2, 3, \dots)$

Open pipes produce all harmonics (both even and odd integers).

2. Closed Organ Pipes (One End Closed)

When one end of the pipe is closed, a displacement node forms at the closed end, while an anti-node remains at the open end. For a pipe of length $L$:

• Fundamental Mode (First Harmonic): $L = \frac{\lambda_1}{4} \implies f_1 = \frac{v}{4L}$
• Third Harmonic (First Overtone): $L = \frac{3\lambda_3}{4} \implies f_3 = \frac{3v}{4L} = 3f_1$
• Fifth Harmonic (Second Overtone): $L = \frac{5\lambda_5}{4} \implies f_5 = \frac{5v}{4L} = 5f_1$
• General Formula ($n$-th Harmonic): $f_n = n f_1 = \frac{n v}{4L} \quad (\text{where } n = 1, 3, 5, 7, \dots)$

Closed pipes produce only odd harmonics.

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