Is it possible to generate a beating wave ?

Giorgis wrote:

Mixing is a multiplicative phenonmenon, not additive. As Don pointed out, you do not get mixing by adding two signals. That can only come from multiplication.

Incidentally, I Googled for a site that would illustrate this, and found this link, which appears to be incorrect:

It states: "When you superimpose two sine waves of different frequencies, you get components at the sum and difference of the two frequencies." This is not true. "Components" is a word typical reserved for superposition. If you superpose two signals x1(t) and x2(t), what you get in a linear medium with unity gain and zero phase is is the sum xs(t) = x1(t) + x2(t). You do not get "components at the sum and difference frequencies". I think the confusion accidentally being promulgated by the physicists has to do with their *mixing* their understanding of physics with their understanding of physiology.

So free space is a linear medium and must abide by the principle of superposition. Two waves flowing in free space must interact in an additive manner, and each must behave in this medium as if the other did not exist.

Consider two waves, S1 and S2, represented by their Poynting vectors:

S1 = E1xH1 S2 = E2xH2

Then, by superposition,

S3 = S1 + S2

and must be equal to E1xH1 + E2xH2. *Additionally* superposition requires that individual E-fields and H-fields add:

E3(r, t) = E1(r, t) + E2(r, t) H3(r, t) = H1(r, t) + H2(r, t)

But, when two signals "beat", it is the result of multiplication. The sum and difference signals that result can almost be seen by eyeballing the definition of cosine using Euler's formula:

's_formula.

I think the so-called "beating" to "generate" different frequencies is a physiological phenonmenon, a human perception. For example, if the ear, which is notoriously non-linear, has the ability to act as an envolope detector

then that would explain why so many people swear that two additive signals produce a different frequency altogether, when they are not: the ear samples the peaks of the multiplicative signal, which is mathematicallly equivalent to the two signals whose frequencies are close.

-Le Chaud Lapin-