Where do "Other" Frequency's Come From when Multiplying Cos Waves?

[CraigH]

When you multiply two cos waves together, where do all the "other" frequency's come f

When you multiply two cos waves with similar frequency's you get a graph that looks like this:

http://www.wolframalpha.com/input/?i=(cos(2*pi*0.8*x))*(cos(2*pi*0.9*x))

With equation y=(cos(2*pi*0.8*x))*(cos(2*pi*0.9*x))

Now if you analyse this wave with a Fourier transform to see the the frequency domain there are lots of other frequency's as well as the 0.8HZ and the 0.9HZ that seem to have come from no where.

Can someone please show me the maths that shows where these frequency's come from?

Thanks

 

[Vargo]

The frequencies you see are .8+.9=1.7 Hz and .9-.8=.1 Hz. See the product to sum trig identities:
http://www.cliffsnotes.com/study_guide/ProductSum-and-SumProduct-Identities.topicArticleId-11658,articleId-11616.html

 

[CraigH]

ahhh okay, thanks, that's helped allot. so just to confirm, when you add 2 waves you get a wave which has 2 frequency's, and these 2 frequency's are the frequency's of the original 2 waves. And when you multiply two waves you use trig identity's to see the equivalent solution as a sum.
Is this correct?

 

[Vargo]

Yes, that is correct (as long as you are talking about sine/cosine waves)