Build the wave equation out of the Maxwell equations and the speed of the wave falls out of the algebra. It comes out as , and the right side holds nothing but the permittivity and the permeability of empty space.
The permittivity fixes how strong an electric field a charge makes, and is measured through the Coulomb law. The permeability fixes how strong a magnetic field a current makes, and is measured from the force between two wires. Neither was determined by looking at light.
Nothing about a medium or a source enters the speed. Only the properties of empty space itself.
Put the two constants in and the answer is about 299.8 million metres per second. The speed of light had already been measured, from astronomy and from experiments on the ground, and the two agreed exactly.
Maxwell did not take that agreement for a coincidence. Light is an electromagnetic wave, and that conclusion follows from here. Electricity, magnetism and light, studied apart until then, closed into one theory.
The two runners in the figure never separate. The point of the picture is that the number from the formula and the number from the measurement are the same number.
In empty space every wavelength travels at the same speed. The three waves in the figure carry wildly different numbers of crests, yet the pattern of each moves at the same rate, and the vertical line keeps cutting all three at the same phase.
Speed, wavelength and frequency are tied by . Since does not move, raising the frequency shrinks the wavelength by the same factor. Fix one and the other is fixed with it.
This is why all the colours of a distant star arrive together. If speed depended on colour, every flicker of the star would reach us smeared out in time, colour by colour, and it does not.
Light runs slower through glass or water than through empty space. The electrons inside are shaken by the wave, those electrons radiate waves of their own, and the new waves add to the original. The sum of all of them is a single wave that lags behind.
The factor by which it lags is the refractive index. For glass it is about 1.5, which puts the speed at roughly two thirds of its value in empty space.
The frequency does not change. If the number of swings per second changed at the boundary the wave would have to tear there. What changes is the wavelength, which shrinks by exactly the factor the speed dropped. That is why the crests crowd together on the right of the figure.