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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!

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

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

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Re-writing the ingtegrand via $\cos(A)\cos(B) = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$ leads to a useful simplification with the answer depending only on $\cos(a)$. –  Dilip Sarwate Oct 18 '11 at 21:00
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