Professor MathTheBeautiful, thank you for Solving Laplace Eigenvalue Problems on a Rectangle in Partial Differential Equations. This is an error free video/lecture on TH-cam TV.
12:45 Can you tell _where_ it has been answered and by _who_ ? [citation needed] heheh ;) 10:30 Well, it _is_ two harmonic series, as you said, and it can be thought of as a metal plate of a xylophone. It produces two harmonic series at the same time, thus the characteristic sound of the xylophone (kinda sounds like two instruments playing at the same time, exactly because of these overlapping harmonic series). Fun fact: I once was driving a car in the night, and there was some metal plate rattling on my control board. It was driving me crazy, so I grabbed it and threw behind on the back seat. I heard its noise when it hit something, and at the same moment the picture of its shape went through my head for a smidgesecond, and I thought to myself: "Hmm.. a tritone, so it's a square plate..." Then it occurred to me what I just did: could it be that I just figured out the shape of the plate by the sound it made?! :o I stopped the car and found the plate on the back seat to take a closer look at it - it was a square indeed :q So started thinking on _how_ I did it, analyzing my train of thoughts, and I figured it out: I heard the musical interval of a tritone, which is in the ratio of √2 to the fundamental tone. The same ratio is between the side of a square and its diagonal! When a square vibrates, there are only two possibilities for the fundamental tone to be produced: either along the side (both sides have the same length, so they produce the same sound in both orthogonal directions) or along the diagonal (either one of the two), in which case it will be √2 times the frequency of the side. If it were a rectangle, I would probably have heard three different tones instead of two.
1) Who proved that X(x)•Y(y) captures all solutions and where can I find this proof? 2) Is it valid for equations with more variables? For example, if u = u(x, y, z, t), would the general solution of Δu = K•∂/∂t(u) be u = X(x)•Y(y)•Z(z)•T(t)?
1) It's not a "theorem". It's just a sentiment, a general guideline. 2). The inner product is a very special thing. It only works for a pair of vectors!
Professor MathTheBeautiful, thank you for Solving Laplace Eigenvalue Problems on a Rectangle in Partial Differential Equations. This is an error free video/lecture on TH-cam TV.
Glad it was helpful!
I'm a musician with interest in mathematics and to me this is pure gold, thank you. So far, your PDE video series is beautiful!
I am waiting for the linear algebra class in which you explain the eigenvalues and why to consider the negative laplacian
Absolutely beautiful.
The echo seems intentional, and well placed.
12:45 Can you tell _where_ it has been answered and by _who_ ? [citation needed] heheh ;)
10:30 Well, it _is_ two harmonic series, as you said, and it can be thought of as a metal plate of a xylophone. It produces two harmonic series at the same time, thus the characteristic sound of the xylophone (kinda sounds like two instruments playing at the same time, exactly because of these overlapping harmonic series).
Fun fact: I once was driving a car in the night, and there was some metal plate rattling on my control board. It was driving me crazy, so I grabbed it and threw behind on the back seat. I heard its noise when it hit something, and at the same moment the picture of its shape went through my head for a smidgesecond, and I thought to myself: "Hmm.. a tritone, so it's a square plate..."
Then it occurred to me what I just did: could it be that I just figured out the shape of the plate by the sound it made?! :o
I stopped the car and found the plate on the back seat to take a closer look at it - it was a square indeed :q
So started thinking on _how_ I did it, analyzing my train of thoughts, and I figured it out: I heard the musical interval of a tritone, which is in the ratio of √2 to the fundamental tone. The same ratio is between the side of a square and its diagonal! When a square vibrates, there are only two possibilities for the fundamental tone to be produced: either along the side (both sides have the same length, so they produce the same sound in both orthogonal directions) or along the diagonal (either one of the two), in which case it will be √2 times the frequency of the side. If it were a rectangle, I would probably have heard three different tones instead of two.
You have some weird echoing sometimes. Are you recording with multiple microphones?
1) Who proved that X(x)•Y(y) captures all solutions and where can I find this proof?
2) Is it valid for equations with more variables? For example, if u = u(x, y, z, t), would the general solution of Δu = K•∂/∂t(u) be u = X(x)•Y(y)•Z(z)•T(t)?
1) It's not a "theorem". It's just a sentiment, a general guideline.
2). The inner product is a very special thing. It only works for a pair of vectors!
Wow, you're quick. Thank you both for the reply and the wonderful lectures!
So can't we guarantee that the spectrum of the rectangle is exacly this? Or these are only SOME eigenvalues?
What's the differences between this and 2d FFT?
Roughly speaking, Fourier transform is concerned with decomposing arbitrary functions with respect to the functions discussed in this video.
What was the Russian joke?
I don't remember, but I'm sure it was hilarious.