Solving the Wave Equation with Separation of Variables... and Guitar String Physics

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  • Опубліковано 4 січ 2025

КОМЕНТАРІ • 84

  • @ehsanulbarihome
    @ehsanulbarihome Місяць тому +5

    Thanks Dr. Brunton. I have finished 3 of your playlist. Vector Calc-PDE is the forth one I am watching. You are very passionate about these math theories. Your explanation are very easy to follow. Just wanted to thank you for your effort. These playlists are going benefit many future generations to come.

    • @Eigensteve
      @Eigensteve  Місяць тому +2

      That is awesome! Thanks for sharing!

  • @fabiofarina9579
    @fabiofarina9579 2 роки тому +23

    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

    • @Eigensteve
      @Eigensteve  2 роки тому +10

      Whoa, that is super cool! I didn't know that

  • @MajdAlmuntaser-b1x
    @MajdAlmuntaser-b1x Рік тому +9

    This guy is incredible. he has helped me so much.Thank you so much

  • @christiancompiles
    @christiancompiles 3 місяці тому +1

    Thank you for the wonderful tie-in with the guitar near the end!

  • @mathhack8647
    @mathhack8647 Рік тому +2

    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,

  • @Tyokok
    @Tyokok Місяць тому +1

    Hi Steve, or anyone, one more question at 28:05 please! You drop sin(c*lambda*t) in G(t) by arguing at Initial Condition when t=0 we have f(x) not zero. But that doesn't give you reason to drop sin(...). Unlike in F(x) you drop cos(lambda*x) because Boundary Condition requires zero so you drop all that's non-zero. And why your original assumption of separation of F(x)G(t) is valid? Any intuition?
    Many thanks if anyone can advice!

  • @sharierkhan7488
    @sharierkhan7488 23 дні тому

    Great..finally I have understood..why we choose -lambda square and how sin or cos vanishes

  • @ravenecho2410
    @ravenecho2410 Рік тому +5

    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

  • @mikebull9047
    @mikebull9047 2 роки тому +10

    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

    • @Eigensteve
      @Eigensteve  2 роки тому +3

      Thanks for letting me know -- always good to know what could be more clear.

    • @ares9748
      @ares9748 Рік тому

      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).

    • @alengm
      @alengm Рік тому +4

      ​@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?

  • @khaledqaraman
    @khaledqaraman Рік тому +1

    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.

  • @ReaganJohnson-n5w
    @ReaganJohnson-n5w Рік тому +1

    Excellent, truly. Thank you for posting.

  • @shakennotstired8392
    @shakennotstired8392 2 роки тому +11

    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).

    • @kingsgambit
      @kingsgambit 2 роки тому +1

      agreed!

    • @郎沛橦
      @郎沛橦 2 роки тому

      agreed! +1

    • @awsomeguy3291
      @awsomeguy3291 Рік тому +1

      Yeah since it's second order we need two I.C's.

    • @khaledqaraman
      @khaledqaraman Рік тому +4

      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.

  • @DaviidReiis
    @DaviidReiis 2 роки тому +10

    TIL: fingers on guitar strings are high-pass filters

  • @nahommerk9493
    @nahommerk9493 3 місяці тому

    At 28:00, I don't think I follow why Steve ignored the sin() part of G just because the Initial condition is equal to zero. I think we need to solve for the coefficient of the sine part of G just like we did for F. Because both G and F have the form 'A*cos() + B*sin()' we, really need 4 givens (2 initial and 2 boundary conditions) instead of 3. I added my own, setting Ut (the time derivative of U at time 0) equal to 0 and then it followed that the coefficient of the sine part of G had to be zero to satisfy that. I think that is the right way to do it... What do you think?

  • @DDDelgado
    @DDDelgado 3 дні тому

    18:00ish the solutions with sin cos or e, are all equivalents. No need to say only cos? A sin and cos are the same with a dephase angle. Its just a matter of taste really.

  • @rakshitjoshi932
    @rakshitjoshi932 2 роки тому +4

    I hope you delve a bit into seismology too :)

  • @faribabiyouki1500
    @faribabiyouki1500 9 місяців тому

    Thank you for the informative video.

  • @inviereno
    @inviereno 18 днів тому

    Hello, thank you for this incredible video! I am a high school student from Korea, and I’m trying to better understand wave equations. I’ve learned that for non-standing waves, the wave function can be expressed as y=Asin(kx−ωt+ϕ). Could you explain how this form of the wave function is derived?
    I might have misunderstood something, so I would really appreciate your explanation. Also, I’m not very confident in my English, so I apologize if my question is unclear. Thank you so much! :D

  • @rohitv1310
    @rohitv1310 Місяць тому

    Thank you!

  • @ruhulhowlader716
    @ruhulhowlader716 6 місяців тому

    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.

  • @rumeysabilcan
    @rumeysabilcan Рік тому

    this video is perfect🥰 thank you so much

  • @Tom-sp3gy
    @Tom-sp3gy 6 місяців тому

    You are the best ever!

  • @mathjitsuteacher
    @mathjitsuteacher 2 роки тому +2

    Hi Steve, the last video you posted was the separation of variables one. I believe you skipped a video.

    • @Eigensteve
      @Eigensteve  2 роки тому +2

      If you go to the "Vector Calculus and PDEs" playlist, they should all be there in order.

  • @alibekyeskermessuly1627
    @alibekyeskermessuly1627 3 місяці тому

    why didn't you use the second initial condition u'(x,0)=g(x)?

  • @rajatsingh-te2wf
    @rajatsingh-te2wf Рік тому +1

    Sir, why are they called eigen values and eigenfunction. Kindly explain. Your small effort will be a great help to me.thanks

    • @Dawlada
      @Dawlada Рік тому

      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.

  • @juancarlossanchezveana1812
    @juancarlossanchezveana1812 9 місяців тому

    Amazing. Thanks

  • @SarjilJawad
    @SarjilJawad 5 місяців тому

    But how is it that we take the constant as -lambda^2

  • @edcoad4930
    @edcoad4930 Рік тому +1

    "resonates" - very good. Comedy aside, great video.

  • @SergeyPopach
    @SergeyPopach 7 місяців тому

    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

  • @Rosalies_
    @Rosalies_ Рік тому

    Really good analysis. Would love a 2D adaptation to emphasize interactions between indices :)

  • @raphaelmoreira1850
    @raphaelmoreira1850 4 місяці тому

    Art.

  • @doc3row
    @doc3row 9 місяців тому

    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 👍

  • @matthewsarsam8920
    @matthewsarsam8920 Рік тому

    wouldn't g(t) have the cos term dropped rather than the sin?

  • @thomasjefferson6225
    @thomasjefferson6225 Рік тому

    Id die of embarsmemt having someone record me playing a guitar lol.😅

  • @pain4743
    @pain4743 6 місяців тому

    Amazing, Than you

  • @McSwagical
    @McSwagical 8 місяців тому +1

    how do they make these videos? does the prof just write backwards???

    • @alexandermuller8858
      @alexandermuller8858 6 місяців тому

      indeed this makes it even more next level. The explanation is in one direction but the writings are backwards

    • @Player-pj9kt
      @Player-pj9kt 15 днів тому

      I think he write normally and then use video editing to flip it so it looks normal to us lol

    • @soumiadje
      @soumiadje 4 дні тому

      I think they actually write on a glass panel, with the camera infront of him, that's why they make the background black and the pen colors bright.

  • @kelvinadimaswijaya9523
    @kelvinadimaswijaya9523 2 роки тому

    12:35 any specific proof of why it's equal to constant?

    • @batu9049
      @batu9049 Рік тому

      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

    • @demr04
      @demr04 Рік тому

      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.

  • @kritb3345
    @kritb3345 2 роки тому

    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...

    • @demr04
      @demr04 Рік тому

      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.

  • @Tyokok
    @Tyokok 2 роки тому

    This is Oh My God, breath taking, amazing lecture!!! NEVER saw a professor explain this clear!!! Hi 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!

    • @demr04
      @demr04 Рік тому +1

      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.

    • @Tyokok
      @Tyokok Рік тому +1

      @@demr04 WOW! clear! Really appreciate it Daniel!

    • @demr04
      @demr04 Рік тому +1

      @@Tyokok your welcome :)

    • @Tyokok
      @Tyokok Рік тому

      @@demr04 this is real fun stuff

    • @demr04
      @demr04 Рік тому +1

      @@Tyokok yeah agree 🤓

  • @الصوتالرخيم
    @الصوتالرخيم 2 роки тому +3

    I wish i have your knowledge

    • @Eigensteve
      @Eigensteve  2 роки тому +3

      Keep watching and you will!

  • @MisterTutor2010
    @MisterTutor2010 11 місяців тому

    Fouier Transform or Series?

  • @ploopsie1403
    @ploopsie1403 Рік тому

    what is Cn?

  • @demr04
    @demr04 Рік тому

    Me costó entender que "buzzcard" se refería a "buscar".

  • @SabahKherfi
    @SabahKherfi Місяць тому

    خسارة كون جاء يفهم بلعربي

  • @AminSatlikh
    @AminSatlikh 2 роки тому

    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!

    • @kelvinadimaswijaya9523
      @kelvinadimaswijaya9523 2 роки тому +3

      well, suggest one then

    • @demr04
      @demr04 Рік тому

      What is the ugly or weak?

    • @AminSatlikh
      @AminSatlikh Рік тому

      @@demr04 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.

    • @declanwk1
      @declanwk1 Рік тому

      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

  • @enginbolat6123
    @enginbolat6123 11 місяців тому

    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)

  • @HosRo4161
    @HosRo4161 Рік тому

    "Harmonics of the planets" is real -- "Kirkwood Gaps" (en.wikipedia.org/wiki/Kirkwood_gap) :)