Taking the time average of the product of two waves

Question

How might one go about calculating the time average of the product of two identical waves with different phases? For example, what would be the time average of:

$$ \cos(k x-w t) \cdot \cos(kx-wt+a) $$

and how would you get it? Thanks!

Answer 1

Your function is fully periodic, which means that your average integral is just an average integral over the fundamental domain.

$$ \begin{align*} \lim_n \frac{\int_0^n cos(kx-wt)cos(kx-wt+a) dt}{n} &= \lim_n \frac{\int_{-n}^n cos(kx-wt)cos(kx-wt+a) dt}{2n}\\ &= \frac{\int_{0}^{2 \pi/w} cos(kx-wt)cos(kx-wt+a) dt}{\frac{2\pi}{w}} \, \end{align*} $$