How Was the Equation v=fλ Derived in Wave Physics?

[Musa Ali]

Hi,

We were recently testing out waves on a string and completing a lab based on it, and I wanted to provide background information on how the equation v=fλ was formulated. I would greatly appreciate it if someone could direct me to a source on it.

Thanks.

 

[Coffee_]

Let me first tell you that in general the equation you wrote down isn't called the wave equation. The wave equation looks like $c^{2}\nabla^{2} f = \frac{\partial^{2} f }{\partial t^{2}}$. Anyway the equation you talk about doesn't really have such a history as you might think - it's very easy to reason. The velocity of something is given by $v=\frac{\Delta x}{\Delta t}$ right? Well let's consider the $\lambda=\Delta x$, what is the corresponding time? It's just what we define to be the period $T$ of the wave, namely, how much times it takes for one wavelength to pass. And so we have $v=\frac{\lambda}{T}$. The fraction $\frac{1}{T}$ corresponds to the frequency $f$ and so we arrive at $v=\lambda f$