From the superposition principle, we can say that in nature, two waves superpose themselves to give a single wave with different (may be more or less than original two) amplitude.

This means addition of two functions (for example, $y=2\sin x$ and $y=2\cos x$) is possible. Similarly, can we see multiplication of two or more functions? If we can, what are the cases?

I have a feeling that resonance can be a result of multiplication of two functions. Can this be true?


closed as unclear what you're asking by sammy gerbil, Jon Custer, Cosmas Zachos, Raskolnikov, Kyle Kanos Dec 19 '17 at 18:56

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    $\begingroup$ Can you be a bit more specific about what you are looking for. My memory of high-school physics is that it mostly involved formulae of the for $A = BC$ and in most cases you can find something else that $B$ and $C$ depend on if you think about it a bit. $\endgroup$ – By Symmetry Dec 14 '17 at 12:00
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    $\begingroup$ But multiplication is simply repeated addition. And resonance is result of the addition of a driving force, as least for simple harmonic motion based systems. $\endgroup$ – user178231 Dec 14 '17 at 12:04
  • $\begingroup$ This post (v2) seems like a list question. See e.g. the method of separation for PDEs. $\endgroup$ – Qmechanic Dec 14 '17 at 12:24
  • $\begingroup$ As you stated you can see the sum of 2 trigonometric functions. It also appears to be the product of two others, by means of trigonometric identities. A phenomenon you could observe to see the multiplication of two functions could be the beat phenomenon when tuning a guitar wikipedia $\endgroup$ – Naptzer Dec 14 '17 at 12:37
  • $\begingroup$ I'm guessing that's a high-school-level question, so answers and many comments are flying a bit too high. If that's correct, my very modest answer might be more in tune... $\endgroup$ – stafusa Dec 15 '17 at 0:12

As you said, the superposition principle allows to add waves. But superposition holds only for linear media, when the response $R$ of the medium to the driving force $F$ is linear with the force, e.g. $R=\alpha F$ with $\alpha$ a proportionality constant.

This is the case for most of optical/electromagnetic phenomena (with $R$ the polarization and $F$ the driving electric field), as well as in acoustic. However, for sufficiently large driving forces, the response is generally not linear anymore so that the response takes a form $R=\alpha F+\beta F^2$. The reason one needs large driving force is that usually $\beta<<\alpha$ and that also explains why we are rarely observing nonlinear effects in everyday life.

Now if F is the sum of two separate excitation $F=F_1+F_2$, the response will be

$R=\alpha F_1+\alpha F_2+\beta F_1^2+\beta F_2^2+2\beta F_1F_2$.

Here you are: the nonlinearity creates a response that is the multiplication of two functions. Note that the superposition principle is apparent only for the term of the 1st power of $F_{1,2}$, but that the cross term $2\beta F_1F_2$ invalidates the superposition principle for the overall result.

In the case of harmonic waves, the square term effectively creates a constant response as well as a component at twice the frequency. This can be see with the trigonometric relation $\cos^2(\omega t)=\frac{1}{2}+\frac{1}{2}\cos(2\omega t)$.

In optics, this property allows the generation of new colors. For example the usual green laser pointers are made of a first infrared laser at a wavelength of 1064nm that passes through a nonlinear crystal that double its frequency, hence dividing its wavelength by two, sot that the output is light at 532nm which is green.

On the other hand the cross term $F_1F_2$ will create frequencies that are the sum and the difference of the frequencies of two possible monochromatic waves interacting in a nonlinear medium. Finally, for even higher intensity, a third order term in the previous response becomes non-negligible, giving rise to another rich variety of phenomena.

A list of effects are described in the following wikipedia page: https://en.wikipedia.org/wiki/Nonlinear_optics


What crosses my mind is any physics where you have two distributions joint together. An example would be Planck's distribution of intensity of black-body radiation & human eye sensitivity (graph below). How we truly see the colors of the sun (their intensities) is given by multiplication of these two distributions.

Sensitivity of human eye. Taken from npl.co.uk

Another naive (I'm not the expert here) example would be conjoint use of multiple filters in a linear circuit/mechanical system. As each filter has a profile, the overall passed signal is multiplication of their profiles.


Can we see multiplication of two functions in nature?

Yes. One example is a pendulum: its oscillations can be described by a sine, and the decreasing amplitude (due to energy losses such as air drag) by an exponential. Therefore, the damped pendulum movement is given by the product of both functions:

$$ \theta(t) = Ae^{- \gamma t} \cdot \sin (\omega \, t). $$

resonance can be a result of multiplication of two functions

Not really. The ideal resonant behavior can indeed be modeled by the product of an oscillating function (such as the sine) and a growing exponential (the one above is a decreasing one). But you ask whether the resonance results from a multiplication, and I don't see how to frame the usual picture is this way.


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