The fast Fourier transform and droplet frequencies

In summary, the conversation discusses the use of FFT (Fast Fourier Transform) in analyzing vibrations of a drop of fluid on a vibrating substrate. The basis functions used in the FFT could be spherical harmonics or Legendre polynomials, depending on the geometry of the system. The FFT can efficiently calculate the amplitudes of each harmonic in the signal. The conversation also touches on the use of FFT in changing from the time domain to the frequency domain. Overall, the participants conclude that FFT is a useful tool in studying these types of vibrations.
  • #1
member 428835
Hi PF!

Suppose we take a drop of fluid and let it sit on a substrate, and then vibrate the substrate. Doing this excites different modes. If someone where to analyze the vibrations, would they take an FFT of the interface, basically reconstructing it from basis functions (harmonics), where the FFT gives the coefficients (magnitude) of each of the basis functions?

Is my understanding correct? I have some followups if so.
 
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  • #2
It seems you are trying to create a physical model of the terms in a 2D FFT.
If this is the case, the drop of fluid will need to be square to reproduce the sine waves that align with the X or Y axis - and you would have to induce vibration with those alignments. The other (diagonal) terms would not be modeled.

Also, what do you mean by "interface"? Is that the top surface of the substrate where it meets the droplet?

Edit:
Looking at your question again, I would say "No".
A circular drop on a vibrating substrate will only have circular modes. They would be simple enough that you would not need to use an Fourier Transform to model them.
 
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  • #3
If you wish to analyze the droplet, the cylindrical geometry would lead to cylindrical Bessel functions and cylindrical harmonics (depending upon your approximations of the system). These are associated with a set of eigenfrequencies and provide a complete basis for representing any initial condition. The structure of the eigenfrequencies is not simple harmonics. A Cartesian FT would seem an inappropriate method..
 
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  • #4
.Scott said:
It seems you are trying to create a physical model of the terms in a 2D FFT.
If this is the case, the drop of fluid will need to be square to reproduce the sine waves that align with the X or Y axis - and you would have to induce vibration with those alignments. The other (diagonal) terms would not be modeled.

Also, what do you mean by "interface"? Is that the top surface of the substrate where it meets the droplet?

Edit:
Looking at your question again, I would say "No".
A circular drop on a vibrating substrate will only have circular modes. They would be simple enough that you would not need to use an Fourier Transform to model them.
By interface I refer to the gas-liquid interface. Reading your edit, wouldn't an FFT work if the basis functions were spherical harmonics?

hutchphd said:
The structure of the eigenfrequencies is not simple harmonics. A Cartesian FT would seem an inappropriate
Okay yes, so an FFT works, but using Legendre (spherical) polynomials as basis functions, right?
 
  • #5
Any Sturm-Liouville problem will generate eigenfunctions that form an orthonormal basis.
I guess I don't know precisely what you mean by an FFT in this context. To me an FT implies a cartesian decomposition using sines and cosines. An FFT implies an efficient calculational technique to obtain same.
So I don't quite understand your question. Please elaborate...
 
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  • #6
hutchphd said:
Any Sturm-Liouville problem will generate eigenfunctions that form an orthonormal basis.
I guess I don't know precisely what you mean by an FFT in this context. To me an FT implies a cartesian decomposition using sines and cosines. An FFT implies an efficient calculational technique to obtain same.
So I don't quite understand your question. Please elaborate...
I think I was uneducated on what an FFT is. I didn't realize its name only refers to rectangular harmonics. Am I correct in understanding that FFTs give the amplitudes for each harmonic when reconstructing a signal?
 
  • #7
I think we have been talking "past" each other. What signal are you trying to ascertain here?
 
  • #8
hutchphd said:
I think we have been talking "past" each other. What signal are you trying to ascertain here?
By "signal" I refer to the droplet vibrations. If equilibrium is a spherical arc, then some disturbance induces surface vibrations. When studying these for this geometry, is it typical for physicists to do an FFT approach, but instead of using sines/cosines, using Legendre polynomials (since they are spherical)?

If not, how would a typical physicist study these?
 
  • #9
Typically the first step in separating the equation is to Fourier Transform (not FFT) in time. The spatial parts of the equation will determine the eigenfrequencies. They will not generally be harmonically spaced and so the response is unique to the geometry.
 
  • #10
joshmccraney said:
I think I was uneducated on what an FFT is. I didn't realize its name only refers to rectangular harmonics. Am I correct in understanding that FFTs give the amplitudes for each harmonic when reconstructing a signal?
The FFT is a digital version of the Fourier Transform function - well suited for computers. It takes an array of ##2^n## samples and the complex result is DFT (Discrete Fourier Transform) of those samples. The array can be of any dimension - linear, 2-d, etc. Each dimension must be ##2^n## samples.

FFT's are often used to change from the time domain to the frequency domain - so the absolute values of each element in the FFT can be used to identify prominent frequencies.
 
  • #12
Thanks everyone! I'll check out the link and appreciate the feedback!
 
  • #13
Hope we helped.. I really like the animations in the article.
 
  • #14
hutchphd said:
Hope we helped.. I really like the animations in the article.
PF always is!

And yea, someone put a ton of time into those.
 

Related to The fast Fourier transform and droplet frequencies

1. What is the fast Fourier transform (FFT)?

The fast Fourier transform (FFT) is an algorithm used to efficiently compute the discrete Fourier transform (DFT) of a sequence or signal. It converts a signal from its original domain (often time or space) to a representation in the frequency domain, allowing for analysis and manipulation of the signal's frequency components.

2. How does the FFT work?

The FFT works by breaking down a signal into smaller sub-signals, applying the DFT to each sub-signal, and then combining the results to obtain the overall DFT of the original signal. This process is much faster than directly computing the DFT of the entire signal, making the FFT a more efficient method for frequency analysis.

3. What are the applications of the FFT?

The FFT has a wide range of applications in various fields such as signal processing, image processing, data compression, and scientific computing. It is commonly used in digital signal processing to analyze and filter signals, in data compression to reduce the size of digital data, and in scientific simulations to solve differential equations and model physical systems.

4. How does the FFT relate to droplet frequencies?

The FFT can be used to analyze the frequency components of a signal, including droplet frequencies. For example, in fluid dynamics, the FFT can be used to analyze the frequency spectrum of droplet sizes in a spray or mist, providing valuable information for studying and optimizing spray patterns and droplet behavior.

5. Are there any limitations to the FFT?

While the FFT is a powerful tool, it does have some limitations. It assumes that the signal is periodic and has a finite length, which may not always be the case in real-world scenarios. Additionally, the FFT is sensitive to noise and can produce inaccurate results if the signal is not properly pre-processed or if there is a large amount of noise present in the signal.

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