Physics

For an open pipe, both ends are antinodes, so the pipe length must fit an integer number of half-wavelengths. The key insight is that the $n^{\text{th}}$ harmonic means $n$ half-wavelengths fit inside length $L$—this constraint directly gives you the wavelength, and frequency follows from the wave equation $v = f\lambda$. Think of it as: longer pipes and lower harmonics produce longer wavelengths and lower frequencies.

For an open pipe, both ends are antinodes, so the pipe length must fit an integer number of half-wavelengths. The key insight is that the $n^{\text{th}}$ harmonic means $n$ half-wavelengths fit inside length $L$—this constraint directly gives you the wavelength, and frequency follows from the wave equation $v = f\lambda$. Think of it as: longer pipes and lower harmonics produce longer wavelengths and lower frequencies.