I The fast Fourier transform and droplet frequencies

[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.

 

[.Scott]

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.

 

[hutchphd]

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..

 

[member 428835]

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?

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?

 

[hutchphd]

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...

 

[member 428835]

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?

 

[hutchphd]

I think we have been talking "past" each other. What signal are you trying to ascertain here?

 

[member 428835]

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?

 

[hutchphd]

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.

 

[.Scott]

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.

 

[hutchphd]

This will have more than you want to know. Ingest slowly and mostly look at the pictures.

https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane#The_axisymmetric_case
And yes the Fast Fourier Transform is a wonderful and computationally remarkably efficient tool.

 

[member 428835]

Thanks everyone! I'll check out the link and appreciate the feedback!

 

[hutchphd]

Hope we helped.. I really like the animations in the article.

 

[member 428835]

Hope we helped.. I really like the animations in the article.

PF always is!

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