Sinc Function in Natural Phenomena

In summary, the function sinc is used in a variety of ways in nature, including as the Fourier transform of aperiodic square waves and as the impulse response of lowpass filters.
  • #1
Bassalisk
947
2
Its my favourite function.

[itex]\Large{\frac{sin(x)}{x}}[/itex]

I first saw it 1 year ago, when we studied limits. I don't know why, but I really like this function.

Can anybody tell me an example, in nature where we have behaviour that has sinc function characteristics.

Why am I asking this?

Fourier transform of aperiodic square wave is this function(give or take few constants).
And this function would be impulse response of lowpass filter.

So a natural process, that can be described with sinc function. Or approximated with it.
 
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  • #2
Hmm, off top of the head if you shine light through a slit it should diffract like sinc function (the amplitude being sinc function squared) for exact same reason why Fourier transform of aperiodic square is sinc.
 
  • #3
Yes yes! How could have I missed that?! The slit experiment. So the slit would be a lowpass filter here, and my light would be a impulse?
 
  • #4
Bassalisk said:
Yes yes! How could have I missed that?! The slit experiment. So the slit would be a lowpass filter here, and my light would be a impulse?
Sort of, though I wouldn't describe it this way.
The intensity is given as integral of the phase from the slit to the screen over the slit width. When you are calculating intensity of the light in direction a (in radians off the centre) you're summing the contributions from every point on the slit multiplied by sines of their phases. The factor varies over the slit as sin(x*sin(a)*2pi/wavelength) (edit: better way to put it is to use complex numbers and e^(i*x*sin(a)*2pi/wavelength) for the multiplier). For a slit that is big relatively to wavelength, the relevant a is small and sin(a)~=a . The summing is same as in Fourier transform, essentially.

You can have some pattern of slits, and then shine laser light through and on the screen see Fourier transform* of that slit pattern (provided that the screen is far enough away). Isn't that cool. Doing the Fourier transform with laser.
*the amplitudes, if you want phase you'll need a beam-splitter to add the beam here so that you can see just the imaginary or just the real part. That's actually the principle behind holography. A hologram is sort of frequency-domain image of the object (not exactly so though).
 
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  • #5
Dmytry said:
Sort of, though I wouldn't describe it this way.
The intensity is given as integral of the phase from the slit to the screen over the slit width. When you are calculating intensity of the light in direction a (in radians off the centre) you're summing the contributions from every point on the slit multiplied by their phases. The phase varies over the slit as sin(x*sin(a)*2pi/wavelength). For a slit that is big relatively to wavelength, the relevant a is small and sin(a)~=a . The summing is same as in Fourier transform, essentially.

You can have some pattern of slits, and then shine laser light through and on the screen see Fourier transform of that slit pattern (provided that the screen is far enough away). Isn't that cool. Doing the Fourier transform with laser.

Interesting. I will research this through. Thanks! Gave me a lot to think about.
 

1. What is the Sinc function and how is it related to nature?

The Sinc function, also known as the sine cardinal function, is a mathematical function that is commonly used in signal processing and physics. It is a periodic function that oscillates between 0 and 1, with a central peak at 0. Its shape is similar to a bell curve and it is used to describe the behavior of waves and vibrations in nature.

2. Can the Sinc function be observed in nature?

Yes, the Sinc function can be observed in a variety of natural phenomena. For example, it is often used to describe the diffraction pattern of light passing through a narrow slit, or the response of a vibrating string or membrane. It can also be seen in the decay of radioactive materials and the distribution of energy in an atom's electron cloud.

3. How is the Sinc function used in scientific research?

The Sinc function is a powerful tool in scientific research as it helps to model and analyze natural phenomena. Its properties allow scientists to understand and predict the behavior of waves and vibrations in various systems. It is also used in digital signal processing to filter out unwanted noise and improve the quality of signals.

4. Are there any real-life applications of the Sinc function?

Yes, the Sinc function has several practical applications in various fields. In engineering, it is used to design filters and control systems. In physics, it is used to study the behavior of waves in different mediums. It is also used in medical imaging, such as MRI scans, to reconstruct images from raw data.

5. Are there any limitations to using the Sinc function in nature?

While the Sinc function is a useful tool in describing natural phenomena, it does have some limitations. It assumes that waves are infinitely long, which is not always the case in real-life situations. It also does not take into account other factors such as interference and non-linear effects. Therefore, it is important for scientists to use the Sinc function in conjunction with other mathematical models to fully understand and predict natural phenomena.

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