harmonics

Meaning: A harmonic is one standing-wave mode of a string or air column, with a frequency that is a whole-number multiple of the fundamental. The fundamental is the first harmonic (n = 1). A mode fits only if the boundaries match: nodes at fixed string ends and closed pipe ends, antinodes at open pipe ends.

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

On the MCAT: Questions give a length and speed and ask for the fundamental or a higher harmonic, or ask how frequency shifts with length, tension, or end conditions. More string tension raises wave speed and so every harmonic, while a longer length lowers them. A spectrum may show peaks at f, 2f, 3f.

Common confusion: Overtone numbering is not harmonic numbering. In strings and open pipes the first overtone is the second harmonic. In a closed pipe the harmonics present are 1, 3, 5, so the first overtone is the third harmonic. A closed pipe's fundamental, v/4L, is half that of an open pipe of the same length.