The wave equations and harmonics
v = f lambdawave speed formulaharmonic seriesstanding wavesbeat frequency
Wave speed, period and frequency, the speed of a wave on a string, and the harmonic series for strings and pipes that decides what note you hear.
Wave Speed (v = fλ)
The universal wave equation: a wave's speed equals its frequency times its wavelength.
Period-Frequency Relation
Period and frequency are reciprocals: seconds per cycle versus cycles per second.
Wave Speed on a String
Waves travel faster on a tighter, lighter string: speed is the square root of tension over linear density.
Fundamental Frequency of a String
Lowest standing-wave frequency of a string fixed at both ends.
Harmonic Frequencies
The nth harmonic of a vibrating system is n times its fundamental frequency.
Fundamental of a Closed Pipe
A pipe closed at one end resonates with a quarter wavelength inside, giving a fundamental of v/4L.
Beat Frequency
Two nearby tones interfere to produce a loudness pulse at their difference frequency, with f₁ the higher of the pair.
How they fit together
v = fλ is the whole of wave kinematics and it applies to every wave there is. The subtlety is which of the three the medium controls: the medium fixes the speed, the source fixes the frequency, and the wavelength is whatever is left over. That is why a sound entering water keeps its pitch but stretches its wavelength, and it is the reason a guitar string's tension and thickness change its note — they change the wave speed on the string, and the length then fixes the wavelength.
For standing waves the question is what the ends do. A string fixed at both ends, and an open pipe, get all harmonics; a pipe closed at one end fits only a quarter wavelength in its fundamental and so produces odd harmonics only, an octave lower than an open pipe of the same length and noticeably hollower — the clarinet against the flute. Use the beat frequency when two close tones interfere: the beat rate is the difference of the two frequencies, which is exactly how a piano is tuned by ear. Common slip: the string wave speed formula wants mass per unit length, not mass.