The equation for beat frequency is as follows: f_b = |f_2 - f_1| Meaning the absolute value of the difference between 2 frequencies. But when you figure out the possiblilities of f_2, you analyze both -f_b and f_b, if you are using the absolute value because negative beat frequencies don't exsist, then why do you analyze the negative beat frequency? You calulate as: f_2 = f_b + f_1 OR f_2 = -f_b + f_1 I don't see why you should analyze the negative beat frequency if it doesn't exsist. Someone please help soon.
You are right that the beat frequency is never negative. The correct way to think about this is best seen through algebra. Start with [itex] f_b = |f_2 - f_1 | [/itex]. Now, remove the absolute signs. In doing so, there are two possibilities. All three frequencies are positive, but f_2 may be larger or smaller than f_1. First case: if f_2 is larger than f_1, in order to geta positive f_b, one gets [itex] f_b = f_2 - f_1 [/itex] Second case: if f_2 is smaller than f_1, in order to *still* get a positive f_b, one must now write [itex] f_b = f_1 - f_2 [/itex] Rearranging gives the two equations you wrote, it's just that the interpretation is different than what you were thinking about. But notice that all three frequencies are always thought as positive. Hope this helps
So then... Oh, I think I understand a bit better now, the || signs are because if the difference is negative, than it still works out to be positive. That was kind of common sense I guess, so then... For solving f_2, you analyze it as either f_2 = f_1 + f_b or f_2 = f_1 - f_b, simply because you don't know whether or not f_2 is bigger or smaller? I think that's right, algebraically you must do this because you don't know which one is bigger, so there must be two possibilites, knowing the beat frequency only tells you the difference between them, so this is both added and subtracted from one of them to account for the two possibilities? Did that sound right? I think i got it now.
Yes, you got it! All three frequencies are always positive. In terms of algrebra, one can do the following: after removing the absolute values, one may write either [itex] f_b = f_2- f_1 [/itex] or [itex] -f_b = f_2 - f_1 [/itex] (notice that f_2 and f_1 are in th same order on the right hand side). But that may give the wrong impression that f_b could be either positive or negative. I prefer to say that after removing the absolute values, one has either [itex] f_b = f_2 - f_1 [/itex] (if f_2 is larger than f_1) or [itex] f_b = f_1 - f_2 [/itex] (if f_1 is larger). Of course, the algebra is the same and people very used to this don't see those two ways as different. But you just have to make sure that you understand what the meaning of the maths is and the second way is probably more clear in terms of the physical interpretation. Patrick