Mechanical waves calculator: strings, sound, Doppler and beats

The relation v = λf holds for every wave, but in mechanical waves it carries a consequence that is easier to miss with light: the speed is a property of the medium, not of the wave. A tighter string carries waves faster, warm air faster than cold. The frequency, by contrast, is imposed by whatever generates the wave and never changes on crossing into another medium. When a sound passes from air into water the frequency stays put, the speed roughly quadruples, and the wavelength quadruples with it.

On a string the speed goes as the square root of tension over linear density. That square root is why tuning an instrument is delicate work: raising the pitch by ten per cent takes twenty per cent more tension, and doubling it would take four times the tension, which the string would not survive. It is also why low notes come from heavier strings rather than from absurdly slack ones.

The Doppler effect and beats are two different ways frequencies combine. Doppler changes the frequency of a single source because it changes the rate at which wavefronts arrive; beats instead come from two stationary sources close in frequency, drifting in and out of phase. The first depends on motion, the second does not — and confusing them is the commonest conceptual slip in the chapter.