A wave that stands still
Pin a rope at both ends and shake it: the wave you send reflects off the far end and interferes with the wave still coming in. At certain frequencies the two combine into a shape that seems frozen — a standing wave. Some points, the nodes, never move at all; halfway between them the antinodes swing with maximum amplitude. The wave no longer travels; it stands and oscillates in place.
Only certain notes fit
Because the fixed ends must be nodes, only waves whose length fits neatly between the ends survive. The string of length L can hold exactly a half-wavelength, or two half-wavelengths, or three, and so on. That quantizes the allowed wavelengths — and therefore the allowed frequencies.
Allowed wavelengths on a string fixed at both ends — n half-wavelengths span the length L.
The frequencies are whole-number multiples of the fundamental f₁: these are the harmonics.
The lowest, n = 1, is the fundamental — the note you hear as the pitch. The higher ones, the harmonics, sound together and give an instrument its timbre (why a violin and a flute playing the same note sound different). A pipe open at both ends behaves like the string — all harmonics — but a pipe closed at one end supports only the odd harmonics, with \lambda = 4L/(2n-1).
Resonance
Why do these special frequencies matter so much? Because of resonance. Drive a system at one of its natural frequencies and each push arrives perfectly timed to add energy, so the amplitude builds enormously. A guitar's body resonates to amplify the strings; a trained singer can shatter a wine glass by matching its resonant frequency; engineers design bridges and buildings to keep their natural frequencies away from wind and earthquakes.
Quick example. A guitar string of length L = 0.65\text{ m} carries waves at v = 400\text{ m/s}. Its fundamental is f_1 = v/(2L) = 400 / 1.30 \approx 308\text{ Hz} — close to the E above middle C. Fret the string (shorten L) and f_1 rises; that is how you play different notes.
How loud? Intensity and decibels
Loudness starts from sound intensity I — the power carried per unit area (in W/m²). A point source spreads its power over an ever-larger sphere, so intensity falls off as the inverse square of distance, I \propto 1/r^2. Move twice as far away and the intensity drops to a quarter.
But the ear responds to enormous ranges — the loudest tolerable sound carries a trillion times the intensity of the faintest audible one. So we compress that range with a logarithm: the decibel scale.
Sound level in decibels, measured against the threshold of hearing I₀.