I mentioned:

> "Perhaps the question to ask -- is it easier to cause a wave machine to slosh by blowing on the surface or to gently rock it?"

This would actually be a very easy experiment to set up. Build a home-made wave machine out of an old aquarium and then compare with two different forcing mechanisms -- (1) with an oscillating translational platform driven by a servo (2) with an oscillating speed fan blowing air over the surface.

It would be easy to measure the average power consumed by each mechanism and see which one takes the least effort to start the wave machine sloshing.

---

The other interesting idea I have is in the analytical realm. I think I have figured out how to automatically extract the principal factors of forcing from a certain class of Mathieu modulated DiffEq's. The key is to create a modulation that is a delta-function train of spikes at a periodic interval. That is an easy convolution to construct as a matrix eigenvalue problem in frequency space. Essentially for every harmonic created one can introduce an additional periodic forcing factor. Solve the matrix of a chosen size and the factors should pop out from the roots. I should get the same answer as the solver finds in the comment above, since the modulation there looks like a train of spikes in the limiting case:

![](http://imageshack.com/a/img922/1014/b5bOdp.png)

If that works, we can get an answer back instantaneously instead of letting the solver grind away finding a miminum error solution. Even if it's not the actual physics involved, it certainly qualifies as an interesting applied math solution.

---

Here is a paper that exactly derives the Mathieu equation for Faraday waves on a sphere.

https://www.pmmh.espci.fr/~laurette/papers/FIS_FA_pub.pdf "Faraday instability on a sphere: Floquet analysis"

I recall seeing a paper from years ago that indicated that the Mathieu equation was inapplicable for a spherical geometry, which may have hindered further research along this path. Interesting that this particular paper is also in the highly regarded Journal of Fluid Mechanics, which recently published a review article from 2016 explaining how to best approach the characterization of wave behavior

[Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited](http://www.unice.fr/rajchenbach/JFM2015.pdf)

This paper includes this jarring statement:

> "For instance, to the best of our knowledge, the dispersion relation (relating angular frequency ω and wavenumber k) of parametrically forced water waves has astonishingly not been explicitly established hitherto. "

This hasn't yet completely sunk in but I still find it strange that at this late date scientists are apparently still struggling to figure out forced wave action in a liquid volume.

The lead author of that paper elsewhere said this:

> The prominent physicist Richard P. Feynman wrote in his cerebrated lectures [10]: *“Water waves that are easily seen by everyone, and which are usually used as an example of waves in elementary courses, are the worst possible example; they have all the complications that waves can have.”* This is precisely these complications that make the richness and interest of water waves. Indeed, despite numerous studies, new waves and new wave

behaviors are still discovered (e.g. , [26, 27]) and wave dynamics is still far from being fully understood.

> "Perhaps the question to ask -- is it easier to cause a wave machine to slosh by blowing on the surface or to gently rock it?"

This would actually be a very easy experiment to set up. Build a home-made wave machine out of an old aquarium and then compare with two different forcing mechanisms -- (1) with an oscillating translational platform driven by a servo (2) with an oscillating speed fan blowing air over the surface.

It would be easy to measure the average power consumed by each mechanism and see which one takes the least effort to start the wave machine sloshing.

---

The other interesting idea I have is in the analytical realm. I think I have figured out how to automatically extract the principal factors of forcing from a certain class of Mathieu modulated DiffEq's. The key is to create a modulation that is a delta-function train of spikes at a periodic interval. That is an easy convolution to construct as a matrix eigenvalue problem in frequency space. Essentially for every harmonic created one can introduce an additional periodic forcing factor. Solve the matrix of a chosen size and the factors should pop out from the roots. I should get the same answer as the solver finds in the comment above, since the modulation there looks like a train of spikes in the limiting case:

![](http://imageshack.com/a/img922/1014/b5bOdp.png)

If that works, we can get an answer back instantaneously instead of letting the solver grind away finding a miminum error solution. Even if it's not the actual physics involved, it certainly qualifies as an interesting applied math solution.

---

Here is a paper that exactly derives the Mathieu equation for Faraday waves on a sphere.

https://www.pmmh.espci.fr/~laurette/papers/FIS_FA_pub.pdf "Faraday instability on a sphere: Floquet analysis"

I recall seeing a paper from years ago that indicated that the Mathieu equation was inapplicable for a spherical geometry, which may have hindered further research along this path. Interesting that this particular paper is also in the highly regarded Journal of Fluid Mechanics, which recently published a review article from 2016 explaining how to best approach the characterization of wave behavior

[Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited](http://www.unice.fr/rajchenbach/JFM2015.pdf)

This paper includes this jarring statement:

> "For instance, to the best of our knowledge, the dispersion relation (relating angular frequency ω and wavenumber k) of parametrically forced water waves has astonishingly not been explicitly established hitherto. "

This hasn't yet completely sunk in but I still find it strange that at this late date scientists are apparently still struggling to figure out forced wave action in a liquid volume.

The lead author of that paper elsewhere said this:

> The prominent physicist Richard P. Feynman wrote in his cerebrated lectures [10]: *“Water waves that are easily seen by everyone, and which are usually used as an example of waves in elementary courses, are the worst possible example; they have all the complications that waves can have.”* This is precisely these complications that make the richness and interest of water waves. Indeed, despite numerous studies, new waves and new wave

behaviors are still discovered (e.g. , [26, 27]) and wave dynamics is still far from being fully understood.