I'm modelling a system that (at the most simple level) is a sum of superimposed waves with different amplitudes. I'll spare all the details but I'd like a nice "rule of thumb" for a beat pattern of multiple waves. The 2-wave beat pattern is obviously nice and easy, but when I was trying to expand for N waves it went a bit complex. I don't need exact, just something to quickly tell if my results are making sense - a small calculation for the order of the beat pattern would be fine (though obviously exact would be better). My frequencies span a range of about 2pi (in units of 1/(units I'm plotting time in)) with a quite even spread so I can't rely on small angle formulae. The number of frequencies changes from a minimum of 5 to about 360 so far (and will probably increase as I progress) so a number independent rule would be nice. Thanks for any replies Happle
What do mean by the number of frequencies exactly? A strict frequency domain analysis would just give you back the frequencies you put in, so how are you defining what constitutes a beat frequency?
The "shape" of the beat pattern will depend on the relative phase of the different components of the signal, as well as their frequency. Try googling for "crest factor" for more information about that.
@mickybob By the "number of frequencies" I mean the number of different waves I'm summing - if I make slight changes in my system, more wavefunctions are relevant and I have to do the same analysis with more functions. My system should have an overall oscillation (according to a journal article I've read) and a time to decohere, and I want to make sure I plot my graph including these ranges. The points on the graph take a really long time to calculate so I can't just plot over a long range and look for it. I'm calling the beat frequency the frequency of this overall oscillation - may not be the right terminology but thought it would suffice. I can probably look at the physics at find an overall frequency for each system but I thought if there was a nice method of obtaining them from the frequencies input it would be easier to include in my program (given the frequencies are already calculated).
Can I get this straight please? Are we dealing with waves (in space) or time varying signals (as received at some point)? If you want to know about 'beats' in a non-linear system, when excited by a number of single frequencies then you would need to specify the actual non linear law (at least to some level). You would expect such a system to produce intermodulation products at frequencies of n1f1+n2f2+n3f3 + etc. where f1, f2, f3 et.c are the input frequencies and n1, n2, n3 etc. are positive, negative integers or zero (i.e all combinations of frequencies). Depending upon the non linear law and assuming it can be reduced to a polynomial, there is a constraint on the values of n, such that n1+n2+n3 + etc will be no greater than the order of the polynomial. For example, if the non linearity is no higher than third order, Ʃ|n| will not be greater than 3, giving possible products at 3f1 or f1+2f2 or 2f3-f2 etc.. The actual levels of the products would depend upon the coefficients of the polynomial law. For many input frequencies, you rapidly get a vast number of possjble combinations. If you are talking about beat patterns with standing waves, then any beats must have wavelengths that are natural modes of the structure.
I'm looking the expectation value of a quantum mechanics operator, measured at a point in time, that is represented as the sum of some waves with associated amplitudes. <P(t)> = constants - [sum(a1*exp(i*w1*t) + a2*exp(i*w2*t) + ... )] The sum is supposed to oscillate before trailing to a fixed point before some artificial affects from only using a finite number of particles in my fermi sea show up (at a later time). I thought if there were a nice way of estimating the oscillation period I could easily decide how many particles I'd have to include and how long in time I'd have to plot over.
It looks like you are looking at one point in space, from the <(P)> equation which is looking at the variation in time. I have no idea how to approach this. You seem to be using some hybrid idea, which is a combination of classical (waves / continuous oscillations) and QM. Is this legit? My previous answer was just giving the standard treatment for finding the effect of a non linearity on multiple frequencies. In a linear system, there are no non linear products (beats). Your "constants" don't look to me as though they constitute a non linearity.
The physics is a bit odd - it's looking at the background of a moving particle in it's rest frame. It came from a Nature journal article so I'm pretty confident it's legit. (If your curious, search for "Quantum flutter of supersonic particles in one-dimensional quantum liquids" it's an interesting paper). The maths is the same as looking at the amplitude of one point in space changing over time, and the sum is all linear. The physics isn't really important to the point (which is why I glossed over it) it's just that I have a mathematical sum of waves with different amplitudes, the waves oscillate at different frequencies in time, and their superposition creates a function that changes in time correspondingly. I basically thought: - The sum of two waves gives a function with two characteristic frequencies, the sum of "N" waves would probably give a function with "N" determining frequencies. In the two wave case with two similar frequencies there f1-f2 determines decoherence showing when the overall function goes to zero and a f1+f2 determining the frequency of the sum in between. Maybe in the "N" wave case there is a frequency (something like the smallest number that can be obtained by summing and subtracting the "N" input frequencies) that determines the main decoherance length, while the other combinations determine the shape of the function in between.
It seems to me that you may be looking for the probability of the sum of the waves exceeding a given value. Would I be right? That could be a bit difficult - on the other hand, it could be pretty straightforward, using simple multiplication of probabilities (?)
It's more that I want to make sure I plot my graph over a large enough range to see all features. I was assuming the sum of multiple waves would have a period that it repeats over, which would give a characteristic frequency (f1-f2 in the simple two wave case). If there were some nice way of estimating this frequency I could then ensure I plot my graph over a range that would show at least one whole oscillation.
I think it may more complicated than just the difference frequency to get the right answer. Although that will give the frequency of the 'beat', it won't tell you the shape of the resultant waveform. If the two (or more) frequencies are harmonically related, then the relative phases will govern whether you get a peaky or ripply resultant, which can strongly affect its maximum value. (Might this not be important to you?) This isn't the case for two or more random frequencies - in fact, for many sources, you get more of a noise-like situation, for which the statistics approach gaussian, I think. I think it's a matter of finding what you actually want out of this. You will probably have realised that I am more into RF and signals than your field. But an alternative view can sometimes show you a way out of a problem.
I was only looking for the frequency of the beat so I could tell my program to stop calculating points outside that range - it takes time to work out each point of the graph and if there is repetition I don't need to spend the time calculating more than one of the subunits. I've got a program that finds all the details of the system - the plot I mention is the explicit sum of all my waves (that have relations between them) so I'm getting all the detail there. I basically was thinking there would be an easy way of putting an upper limit on the time to repeat knowing the frequencies input, so I could cut off the graph before repetition.
You won't know about the detailed repeat time of a particular beat pattern amplitude, as I said, because that will depend on the actual frequencies and the phase relationship. If you don't care about that particularly then what may be enough for you will be to find the lowest possible resultant frequency that you can get by using different combinations of + and - n times all the frequencies you are injecting. That will give you the lowest beat frequency and the time you want, I think. Suitable choice of the injected frequencies could shorten this time. PS If you had said what you said in the last post, I think we could have come up with this idea much sooner.
Thanks - I do tend to use a phrasing that no-one else would (not to mention sometimes creating my own terminology that is in direct contradiction with the standard) I'll work on it
It looks like it will - unfortunately my supervisor came round yesterday and asked me to check my values against something I hadn't thought of and it turns out I was getting the wrong values anyway (always to way :grumpy:). I've been trying to find the fault since yesterday so haven't had a chance to use it.