Two Physics Questions - Help is greatly appreciated :)?

Question

QUESTION ONE:

Is there any way the two individual frequencies can be combined to give the same beat frequency previously measured?

QUESTION TWO:

There are commercial products available called active noise cancellers, which consist of a set of headphones, microphones, and some electronics. Intended for wearing in noisy environments where the user must still be able to hear (for example, radio communications), the headphones reduce noise far beyond the simple acoustic isolation of the headphones. How does such a product work?

You don't have to answer both, but whatever you can help with is greatly appreciated (I will choose best answer).

Thanks everyone!

Answer 1

Question ONE:

Beat frequency a = frequency 1 minus frequency 2
Beat frequency b = frequency 2 minus frequency 1

Question TWO:

There are microphones inside the closed cups of the earphones. A signal processor does a lot of calculations/operations but basically tries to solve the problem in such a way that the difference from the microphones inside the headphone in respect to the audio signal coming from the audio source to the headphone-electronics is zero. Like that the noise is canceled.

Answer 2

QUESTION ONE:
sin(ω1t) + sin(ω2t) =
2sin((ω1 + ω2)t/2)cos((ω1 - ω2)t/2)

QUESTION TWO:
The headsets have an amplifier which gives unity amplification at 180° phase difference to the original input at virtually all frequencies. There is a mixer circuit to add in signal from a second source.

Answer 3

Sorry in your good fortune, however Bill did not fairly reply #a million, and his reply for #two is flawed. For two, he used diameter alternatively of radius. Just plug in therefore, will have to determine.