0
$\begingroup$

I have a product of two cosine waves, each with a different period.

$$f(x) = \cos(\omega_1t-\delta)\cos(\omega_2t-\delta)$$

I would like to find the time average of this product using:

$$\frac1T \int_0^T \cos(\omega_1t-\delta)cos(\omega_2t-\delta)dt$$

The period of $f(x)$ is $T = \omega_1\omega_2$, but with this limit, I get a computer solution that seems too complicated to be correct. Is there any reason to believe this integral has a palatable solution? If the integral is analytic, please sketch the method of solution.

$\endgroup$
1
$\begingroup$

Hint: You may solve it by using$$\cos A.\cos B=\frac{1}{2}(\cos(A+B)+\cos(A-B))$$

Thus: $$\frac{1}{2T}\int^T_0 \cos((\omega_1+\omega_2)t-2\delta)+cos(\omega_1-\omega_2)t.dt $$

Hope you can solve further?

$\endgroup$
  • 1
    $\begingroup$ I should have remembered that identity. The integral is trivial from here, thank you. The result is uglier than I expected, but such is life... $\endgroup$ – B. Bush Apr 30 '17 at 0:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.