I have learned that to construct a wave pulse, we need to superimpose multiple sine waves of different frequencies which all interfere to produce the pulse. What I don't understand is that a wave pulse is produced at a certain place and time, i.e. a laser that is quickly turned on and off at a given position at a given time, yet the sine waves that superimpose and make up the pulse extend over all space and time to infinity, including before the laser was even turned on. If a wave pulse is really made up of these sine waves, how could these waves have been produced before the pulse even existed?
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asked Jul 11, 2016 at 13:47
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2$\begingroup$ a wave pulse can be mathematically created by the superposition of sine waves. $\endgroup$– LelouchCommented Jul 11, 2016 at 13:54
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4$\begingroup$ It straightforwardly follows from the fact that $e^{ikx}$ is a basis of $L^2([a,b])$, therefore any $L^2$ function can be written as (infinite) expansion of sine and cosine with some coefficients. $\endgroup$– gentedCommented Jul 11, 2016 at 13:58
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2$\begingroup$ For most students this is the first time they encounter the idea of a basis in a context other then simple 3-space. That's why most books introduce the idea with some set of images to show the increasingly good approximation as terms are added. But it works in much the same way that a linear combination of unit vectors in the $x$, $y$, and $z$ direction can be combined linearly to form any vector in ordinary 3-dimensional Cartesian space. $\endgroup$– dmckee --- ex-moderator kittenCommented Jul 11, 2016 at 14:10
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$\begingroup$ @GennaroTedesco I'm guessing the OP wants a lower level than that and indeed is probably asking for an intuitive motivation for that nontrivial theorem. Although he/she does list Fourier-Transform as a tag - it's a little unclear. $\endgroup$– Selene RoutleyCommented Jul 11, 2016 at 14:39
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1$\begingroup$ Just to clarify, what's being asked about here appears to be what is technically called a pulse wave or a pulse waveform, which may be thought of as a periodic rectangular wave. When used alone, a pulse refers to a one-time aperiodic event. And, AFAIK, wave pulse is ambiguous term. The distinction is important because FFTs model periodic waveforms, aperiodic wavefroms require a different approach. $\endgroup$– RBarryYoungCommented Jul 11, 2016 at 19:45
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3 Answers
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In this case it's probably best to be pragmatic. A pulse can be described as a superposition of sine waves that extend infinitely into space and time. But it's just that: a mathematical description that is useful for your purposes. There is not necessarily a physical meaning connected to it. Nevertheless, in quantum mechanics the wave-description of phenomena has turned out to be incredibly successful. But it can often lead to some confusion or counter-intuitive results.
If you model a short laser pulse as a superposition of infinitely many sine waves, you also run into other problems than the ones you described. For example, you will discover that the phase velocity (the velocity of the sine wave trains) will be higher than the speed of light, which is apparently clashing with general relativity that prohibits faster-than-light propagation of information.
Again, the solution of this apparent paradox is that the sine waves do not carry physical meaning. However, the group velocity (the velocity of the pulse itself, constructed from these sine waves) can only move at the speed of light or below.
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I just wanted to add to a previous (very accurate) answer: you can think of it as an Fourier expansion of the actual (physical) wave profile. It is not a real life process, it is a mathematical approximation. The wave pulse can be thought of as a superposition of plane waves, which happens to interfere destructively in entire space, except for the localized region - location of the pulse.
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$\begingroup$ So, this means, we could take some other mathematical function, something completely different from a sine, to fit onto the actual wave, it's just that the sine function is among the most convenient and easiest to work with? (some image matching algorithms use cosines to convert to the frequency domain, as they are a better fit for the usual patterns we are interested in) $\endgroup$– vszCommented Jul 12, 2016 at 6:27
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$\begingroup$ @vsz well, if you are not concerned with parity, it doesn't matter whether you expand in sin or cos. Yes, essentially you can use any orthonormal set of functions as an expansion basis. One of the reasons, why ppl use sin and cos is because they are the solutions to harmonic oscillator (one of the only problems we know how to solve exactly and understand pretty well). $\endgroup$– MsTaisCommented Jul 12, 2016 at 18:19
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This is a lot more subtle problem than is indicated in any of the comments. The problem is not just the issue of how the sum of non-causal signals can approximate a causal one, but how is it possible that while all real-life signals must start and stop at some time they must also be band-limited beyond some frequency, but as we know these two are contradictory (Paley-Wiener). The resolution of this paradox along with the present question is in Slepian: On Bandwidth, Proc. IEEE, 1976 vol. 64, No. 3 pp292-300. In short, the answer lies in dual approximation valid in both time and frequency domains, and in what we may call a legitimate mathematical model of a physical signal. Slepian's article is very readable and can be understood without much mathematical baggage.
answered Jul 11, 2016 at 19:50
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$\begingroup$ This is the best answer, even though may well be beyond the scope the OP intends. $\endgroup$– HansCommented Jul 12, 2016 at 2:43