harmonic number

Meaning: A string fixed at both ends or a pipe open at both ends supports standing waves only at certain wavelengths. The harmonic number n counts the half-wavelengths that fit in the length L, so n = 1 is the fundamental (first harmonic), n = 2 the second harmonic, and so on. Each harmonic has a frequency n times the fundamental. For a pipe closed at one end, which has a node at the closed end and an antinode at the open end, only odd values n = 1, 3, 5, … fit.

Equation: fn = nv/(2L) for a string fixed at both ends or a pipe open at both ends, with n = 1, 2, 3, …. For a pipe closed at one end, fn = nv/(4L) with n = 1, 3, 5, …. Here fn is the frequency in Hz, n is the dimensionless harmonic number, v is the wave speed in m/s, and L is the length in m. Equivalently, λn = 2L/n for the first case.

On the MCAT: Passages describe a guitar string, organ pipe or resonance tube. You may be asked for the frequency or wavelength of the third harmonic, the ratio of two harmonics, or how the fundamental changes when tension (v = √(T/μ)) or length changes.

Common confusion: Harmonic and overtone are numbered differently. The first overtone is the second harmonic in a string or open pipe, but the third harmonic in a closed pipe. Also, n is not the number of nodes: a fixed string has n − 1 nodes between its ends.