Solving the Wave Equation with Separation of Variables... and Guitar String Physics
HTML-код
- Опубликовано: 22 июл 2024
- This video explores how to solve the Wave Equation with separation of variables. This is a cornerstone of physics, from optics to acoustics, and we use the physics of guitar strings to make this intuitive.
@eigensteve on Twitter
eigensteve.com
databookuw.com
This video was produced at the University of Washington
%%% CHAPTERS %%%
0:00 Introduction
5:14 Initial Conditions and Boundary Conditions for the Wave Equation
8:40 Separation of Variables
16:24 Solving the ODEs for Space and Time
29:25 General Solution of the Wave Equation
32:24 Recap
34:50 Guitar String Physics
41:06 Method of Characteristics 
Наука
Fun fact about history of music and science. Equal temperament, the way we divide octaves in notes in multiple log_2(1/12) was rediscovered in mid1500 by Vincenzo Galilei. He's Galileo father
Whoa, that is super cool! I didn't know that
Excellent, truly. Thank you for posting.
This guy is incredible. he has helped me so much.Thank you so much
Ce valeureux Professeur est génial, il a le don d'enseigner et de simplifier les concepts qu'on prenait parfois pour des citadelles impénétrables . Un grand Merci pour vous cher Monsieur . may God Bless you , I know it's hard, but. you have to publish more for the best of your thirsty and faithful audience, . Thanks,
Thank you for the informative video.
this video is perfect🥰 thank you so much
for the negative sign, similar to the heat equation video, diffision was negative bc it was the state returning to equillibrium (exuding heat to the env), similarly the string will be returning to equillibrium in a non-preturbed state (at rest)
at least kinda how i think of it, might help others with sign of lambdas
I like this way of thinking.
Frequency: number of waves passing by a specific point per second. Period: time it takes for one wave cycle to complete. The relation between frequency (f) and time period (T) is given by f=1/T. Notice that (f) increases when L is shortened.
Really good analysis. Would love a 2D adaptation to emphasize interactions between indices :)
Amazing. Thanks
You are the best ever!
the step to eliminate the sin solution part is not clear. and the constant c is employed twice in 2 different uses-
But that's nitpicking. great lecture
Thanks for letting me know -- always good to know what could be more clear.
He removed the sin part because sin(c£t) when t= 0 is equal to zero.
Sin (0) = 0 . So we removed it .
Because according to initial condition when t= 0 , U(x,0) = f(x).
@ares9748 that just means that sin in G doesn't contribute to u at t=0. It still doesn't contradict the initial condition, so why remove it?
TIL: fingers on guitar strings are high-pass filters
Amazing, Than you
Maybe the sin term in the general solution for G(t) should not have been dropped off? the coefficient associated with that term will be determined by a 2nd initial condition, i.e., u"(x,0).
agreed!
agreed! +1
Yeah since it's second order we need two I.C's.
At 28:08 he assumed implicitly that dU/dt (x,0) =0 which means the initial velocity is zero. So that's an extra initial condition that was not mentioned at the beginning.
Professor please show me that when a unit mass as a wave propagate and transfer energy to the mass energy is kept constant. I can find particle velocity and shear strain for a shear wave and the displacement at a particular point for any time t but I don’t get the total energy of at the point does not main the same value. As shear strain is directly related to the particle velocity, is it that I have to consider either particle velocity or shear strain plus displacement related velocity in the perpendicular direction of displacement. Please help me.
wouldn't g(t) have the cos term dropped rather than the sin?
I hope you delve a bit into seismology too :)
Would lambda be the eigen vectors and Bn be the eigen values? When I imagine an infinite sum of frequencies forming a solution, I think of each frequency as the eigen vector and Bn is the correct weight. I may be confusing eigen vectors for Fourier basis functions...
A linear combination of eigenvector don't need to be weigthed by its eigenvalues. In this case, the sines are eigenvector or "eigenfunction" of the differential operator, lambdas are the eigenvalues, and the Bs are the unique weights that can form the initial distribution with the fourier series.
"resonates" - very good. Comedy aside, great video.
it turned out to be that we got a vector space with orthonormal basis of infinite dimension that has infinite amount eigenfunctions and their corresponding eigenvalues… just like in quantum physics
Sir, why are they called eigen values and eigenfunction. Kindly explain. Your small effort will be a great help to me.thanks
I would recommend you to refer Linear algebra to understand that point. Once you understand eigenvalues it will be easy to understand eigenfunction. It is a bit tough but very beautiful.
Newton wanted to apply music theory to his prism spectrum. He could "see" 6 colours. Red orange yellow green blue and the darker blue that he called Violet. But diatonic scale A-G is 7 notes. So he invented "indigo" to appear between blue and violet. Musical string analogy achieved 👍
Hi Steve, the last video you posted was the separation of variables one. I believe you skipped a video.
If you go to the "Vector Calculus and PDEs" playlist, they should all be there in order.
Id die of embarsmemt having someone record me playing a guitar lol.😅
Steve, why you call lambda square Eigenvalue? How does this relate to matrix Eigenvalue? Thank you so much again for such vivid elegant explanation of wave equation video!
If you think of a differential operator D, applying to a function and setting a eigenvalue problem is:
D(y) = a*y
where "a" is a scalar and "y" is a real-value function. Solving for "y" gives y=e^(ax), so you can see that e^(ax) is an eigenvector or "eigenfunction", meanwhile "a" is it's eigen value.
In this case, the eigenvalues are infinitly many because it's a partial differential equation, meaning that it's has infinite solution. In a normal ODE, has finite many of them, so there is finite quantity of solutions.
@@rajinfootonchuriquen WOW! clear! Really appreciate it Daniel!
@@Tyokok your welcome :)
@@rajinfootonchuriquen this is real fun stuff
@@Tyokok yeah agree 🤓
12:35 any specific proof of why it's equal to constant?
hey hello it not need proof that space cant equal time at there like 5x is not equal to t or 5t or something it just can be if they equal a constant
The only function which can acept non related argument is the constant function, because the other case is for any f, g: R to R such that f(x) = g(y), means that x = f^-1(g(y)) or viceversa, which can't be because x and y are not related by any function.
how do they make these videos? does the prof just write backwards???
indeed this makes it even more next level. The explanation is in one direction but the writings are backwards
Fouier Transform or Series?
I thought the same
Me costó entender que "buzzcard" se refería a "buscar".
what is Cn?
30:12
@@lt4376 thanks!
I wish i have your knowledge
Keep watching and you will!
The solution of wave eq. is too ugly here and it presented in a weak way. There are far better and cleaner ways of defining the solution analytically! Such a pity!
well, suggest one then
What is the ugly or weak?
@@rajinfootonchuriquen The way of presenting the solution in comparison with others who did the same. Up to this point, almost everything was smooth and pretty. I think he needs to improve it.
this is a brilliant presentation by a master teacher. He has put so much work into it and then gives it to the community for free. He deserves our respect
Can you solve this question? I couldn't solve it. Can you help me?
Find the distribution 𝑢(𝑥, 𝑡) by writing the wave equation and boundary conditions for a rod (one dimension) of length L=1 unit, with both ends fixed and whose initial displacement is given by 𝑓(𝑥), whose initial velocity is equal to zero. (𝑐2 = 1, 𝑘= 0.01)
𝑓(𝑥) =ksin(3𝜋x)
"Harmonics of the planets" is real -- "Kirkwood Gaps" (en.wikipedia.org/wiki/Kirkwood_gap) :)