# Taking the time average of the product of two waves

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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\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*}
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