In this section, we intend to look at the Wave function on our best friend Mathematics perspective. Shout out to Prof. Mittleman and his team for their amazing slides!
https://www.brown.edu/research/labs/mittleman/sites/brown.edu.research.labs.mittleman/files/uploads/lecture02_0.pdf
If you studied the slides above, you would probably have some senses on the following context.
We can derive the wave equation from Maxwell’s equations. Here it is, in its one-dimensional form for scalar (i.e., non-vector) functions $f$.
$$ \dfrac{\partial^{2}f}{\partial x^2}-\dfrac{1}{v^{2}}\dfrac{\partial^{2}f}{\partial t^{2}} = 0 $$
where $f$ stands for amplitude of the wave, $x$ is the dimensional that the wave is propagating and $t$ represents the time.