the wave speed

When a wave travels, how fast does it go? That single number is the wave speed, written c. It is the constant sitting in front of u_xx in the wave equation u_tt = c^2 u_xx, and it tells you how many metres the disturbance covers each second.

Crucially, c is fixed by the medium, not by the wave. For a string c = sqrt(T/rho), where T is the tension and rho the mass per length — tighten the string or make it lighter and waves run faster. For sound in air c is about 343 m/s, set by the air's pressure and density; for light in vacuum c is the famous 299,792,458 m/s. In the equation, c always appears as c^2 (the speed squared), which is why only its magnitude matters and a wave is equally happy travelling left or right. Where does c come from in the equation? The travelling shape f(x - ct) satisfies the wave equation exactly: differentiate twice in t and you get c^2 f'', differentiate twice in x and you get f''; the c^2 makes the two sides match, so c is precisely the speed at which the profile slides along.

Two warnings. First, in a simple (non-dispersive) wave equation a single c carries every wavelength at the same speed, so a pulse keeps its shape; in a dispersive medium different wavelengths travel at different speeds and a pulse spreads — then one must distinguish phase speed from group speed, and a single c no longer tells the whole story. Second, c is a property of the equation's coefficients; if the medium changes (the string suddenly gets heavier, light enters glass), c changes, and that jump is exactly what causes reflection and transmission.