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AM electromagnetic waves: 20 KHz modulation frequency on an astronomically-low carrier frequency
Jim Kelley wrote
Hein ten Horn wrote: That's a misunderstanding. A vibrating element here (such as a cubic micrometre of matter) experiences different changing forces. Yet the element cannot follow all of them at the same time. As a matter of fact the resulting force (the resultant) is fully determining the change of the velocity (vector) of the element. The resulting force on our element is changing at the frequency of 222 Hz, so the matter is vibrating at the one and only 222 Hz. Under the stated conditions there is no sine wave oscillating at 222 Hz. The wave has a complex shape and contains spectral components at two distinct frequencies (neither of which is 222Hz). Not a pure sine oscillation (rather than wave), but a near sine oscillation at an exact period of 1/222 s. The closer the source frequenties, the better the sine fits a pure sine. Thus if you wish to get a sufficient near harmonic oscillation, conditions like "slow changing envelope" are essential. It might be correct to say that matter is vibrating at an average, or effective frequency of 222 Hz. No, it is correct. A particle cannot follow two different harmonic oscillations (220 Hz and 224 Hz) at the same time. The particle also does not average the two frequencies. Hmm, let's examine this. From the two composing oscillations you get the overall displacement: y(t) = sin(2 pi 220 t) + sin(2 pi 224 t) From the points of intersection of y(t) at the time-axes you can find the period of the function, so examine when y(t) = 0. sin(2 pi 220 t) + sin(2 pi 224 t) = 0 (..) (Assuming you can do the math.) (..) The solutions a t = k/(220+224) with k = 0, 1, 2, 3, etc. so the time between two successive intersections is Dt = 1/(220+224) s. With two intersections per period, the period is twice as large, thus T = 2/(220+224) s, hence the frequency is f = (220+224)/2 = 222 Hz, which is the arithmetic average of both composing frequencies. The waveform which results from the sum of two pure sine waves is not a pure sine wave, and therefore cannot be accurately described at any single frequency. As seen above, the particle oscillates (or vibrates) at 222 Hz. Since the oscillation is non-harmonic (not a pure sine), it needs several harmonic oscillations (frequencies, here 220 Hz and 224 Hz) to compose the oscillation at 222 Hz. Obviously. It's a very simple matter to verify this by experiment. Indeed, it is. But watch out for misinterpretations of the measuring results! For example, if a spectrum analyzer, being fed with the 222 Hz signal, shows that the signal can be composed from a 220 Hz and a 224 Hz signal, then that won't mean the matter is actually vibrating at those frequencies. :-) Matter would move in the same way the sound pressure wave does, To be precise, this is nonsense, but I suspect you're trying to state somewhat else, and since I'm not able to read your mind today, I skip that part. the amplitude of which is easily plotted versus time using Mathematica, Mathcad, Sigma Plot, and even Excel. I think you should still give that a try. No peculiarities found. gr, Hein |
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