# Who wrote the equation $c=\lambda\nu$ for the first time?

I could not find who wrote the equation $$c=\lambda\nu$$ for the first time. Neither I found a name for this equation. A user from Physics forum thinks it is too obvious for anyone familiar with wavelength and frequency, and once light has been understood traveling as electromagnetic waves, someone must has described it with this equation. Does anyone know anything more about it?

• Hi Mauricio, maybe. I just wondered about this equation in particular, since its simplicity is charming and wondered whether it can be applied also to G since both are waves at same speed. Mar 24 at 13:04

Relations between the frequency and length of a string are known since Antiquity, but a more precise mathematical treatment would have to wait until the invention of calculus.

The wave equation was first studied by Brook Taylor (1713) and later by Jean d'Alembert (1746). (As for who really did it first check this answer: Who discovered the wave equation?). I was not able to find the explicit equation from commentary of their work but Jakob Hermann in Phoronomia (a commentary on the work of Taylor and Newton) in 1716 writes:

$$V=\lambda/T$$

where $$V$$ is the speed of sound and $$T$$ the period of oscillation. The rest of the analysis of Hermann was wrong. Source: The Evolution of Dynamics: Vibration Theory from 1687 to 1742 pp

The solutions of the wave equation are of the form $$f( x/\lambda-\nu t)$$ (in the argument you have $$x$$ position, $$\lambda$$ wavelength, $$\nu$$ frequency and time $$t$$), which can also be written as another function $$g(x-ct)$$ (where $$c$$ is the speed of the wave). For the simplest case (describing sinusoidal waves), it leads to your formula $$c=\nu\lambda$$ (known as linear dispersion relation).

As for light. Thomas Young showed definitely in 1800 that light behaves like a wave using a double slit experiment. Experiments showed that light in air seemed to follow that simple linear relation $$c=\nu\lambda$$ when it is not travelling inside a media.

In 1855, Wilhelm Weber and Rudolf Kohlrausch suggested that the speed of light was equivalent to the ratio of electric and magnetic constants. In 1864, James Clerk Maxwell, compiled all the different equations of electromagnetism and finally showed that electromagnetic fields propagate as waves, and in vacuum these waves follow $$c=\nu\lambda$$ where $$c$$ was the calculated to be equal to the experimental value for the speed of light. So visible light is a type of electromagnetic wave with linear dispersion.

In the comments you mentioned gravitational waves. In 1915 Albert Einstein published his theory of general relativity (suggesting that gravity is related to the curvature of space-time), and in 1916 he derived the equations for the gravitational waves. These equations are very complex and describe the distorsion of spacetime, the same arguments that one needs to write the equation. However for very simple spacetimes geometries it also follows $$c=\nu\lambda$$, where $$c$$ is the speed of light. The fact that for light and gravitational waves the speed is the same is related to relativity, where $$c$$ is the maximum speed a wave can travel and it's related to causality.

• Thanks Mauricio. It appears much clearer to me now. Mar 26 at 7:22