Longitudinal stationary waves in wind instruments

In wind instruments, a vibrating source at one end sets up longitudinal stationary sound waves within the air column. Areas of compression and rarefaction form at fixed positions.

For an open pipe:

• Antinodes occur at the open ends.
• The lowest frequency is produced when length l = λ/2.

For a closed pipe:

• A node occurs at the closed end.
• The lowest frequency has l = λ/4.

Conditions for resonance and stationary waves

Resonance occurs when:

• The frequency matches a natural frequency of the air column.
• Constructive interference produces stationary waves.

The pipe length dictates which wavelengths, and hence frequencies, can resonate.

Relating length and wavelength to resonant frequency

For an open pipe with length l:

• Minimum l = λ/2 for the 1st harmonic
• Higher harmonics have l = λ, 3λ/2 etc.

For a closed pipe with length l:

• Minimum l = λ/4 for the 1st harmonic
• Higher harmonics have l = 3λ/4, 5λ/4 etc.

Since v = fλ, shorter lengths produce higher resonant frequencies.