harmonic

Meaning: A harmonic is a standing-wave pattern a bounded medium, such as a string or air column, can sustain. The first harmonic (n = 1) is the fundamental, the lowest frequency; the nth harmonic has n times that frequency. A string fixed at both ends has a node at each end, and a pipe open at both ends has an antinode at each end; both allow every integer n. A pipe closed at one end has a node at the closed end and an antinode at the open end, so it allows odd n.

Equation: fn = nv/(2L) for a string or open pipe, n = 1, 2, 3, …; fn = nv/(4L) for a pipe closed at one end, n = 1, 3, 5, … Here fn is frequency (Hz), v is wave speed (m/s), L is length (m), and n is a dimensionless integer.

On the MCAT: Passages use instruments or tubes. Find the third harmonic from the fundamental, predict pitch when length shortens or tension rises (v = √(T/μ) on a string), or compare open and closed pipes of equal length; the closed pipe's fundamental is half as high.

Common confusion: Overtones are numbered from the second harmonic. The first overtone is the second harmonic on a string or open pipe, but the third harmonic in a closed pipe.