Phase Velocity versus Group Velocity: Wave Dispersion

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  • Опубликовано: 8 фев 2025

Комментарии • 332

  • @EugeneKhutoryansky
    @EugeneKhutoryansky  5 лет назад +15

    To see subtitles in other languages: Click on the gear symbol under the video, then click on "subtitles." Then select the language (You may need to scroll up and down to see all the languages available).
    --To change subtitle appearance: Scroll to the top of the language selection window and click "options." In the options window you can, for example, choose a different font color and background color, and set the "background opacity" to 100% to help make the subtitles more readable.
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    --If you believe that the translation in the subtitles can be improved, please send me an email.

  • @pantherplatform
    @pantherplatform 5 лет назад +449

    I play these videos on the big screen in the waiting room at my dental office so my patients forget all about the terror that awaits them.

    • @HiAdrian
      @HiAdrian 5 лет назад +12

      That is cool man!

    • @BlackEyedGhost0
      @BlackEyedGhost0 5 лет назад +30

      Making the world smarter, one patient at a time.

    • @RadioReprised
      @RadioReprised 5 лет назад +14

      I practice the Standing Wave form at the Doctor's Office.......I stand up and wave goodbye before the appointment!

    • @efeplevneli9063
      @efeplevneli9063 5 лет назад +1

      Haha

    • @pantherplatform
      @pantherplatform 5 лет назад

      @@RadioReprised that's the best.

  • @BariumCobaltNitrog3n
    @BariumCobaltNitrog3n 5 лет назад +83

    I like to wave at people. Even better is when they wave back. I love that.

    • @germanvisitor2
      @germanvisitor2 5 лет назад +10

      If you wave at each other at the same speed you cannot move because it is a standing wave.

    • @BariumCobaltNitrog3n
      @BariumCobaltNitrog3n 5 лет назад +2

      @@germanvisitor2 I thought a standing wave is when you stop walking to wave so they can see you better.

    • @JB-or9yw
      @JB-or9yw 4 года назад +1

      Me two! And scientifically, that is plausible lol

    • @JB-or9yw
      @JB-or9yw 4 года назад +1

      @@germanvisitor2 maybe if you move your observation point relative to your hands lol

  • @SolveEtCoagula93
    @SolveEtCoagula93 5 лет назад +115

    Oh how I wish I could have seen that simple animation of phase vs group when I was an undergraduate!

    • @millieh3179
      @millieh3179 4 года назад +12

      Undergrad here, can confirm it is very useful!

    • @SolveEtCoagula93
      @SolveEtCoagula93 4 года назад +5

      @@millieh3179 Even more jealous now! Glad that it is helping you though. Good luck with your course Millie.

    • @yashas9974
      @yashas9974 4 года назад +1

      @@millieh3179 Ditto.

    • @davel7037
      @davel7037 4 года назад

      Almost managed to understand this through though process, then i came across this video which cleared further doubts.

    • @jaiho2623
      @jaiho2623 3 года назад +1

      Hey neil, you are sooooo correct.

  • @jaiho2623
    @jaiho2623 3 года назад +5

    Last 5 seconds explained to me more about "Standing Waves" then entire books!!!

  • @markostojiljkovic7100
    @markostojiljkovic7100 5 лет назад +8

    I knew about group velocity... now i understand it. You couldn't explain it better than this! Great job!

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  5 лет назад

      Thanks for the compliment.

    • @WojciechowskaAnna
      @WojciechowskaAnna Год назад

      group velocity is parital derivative of angular frequency over wave number, without undurstanding it, i don't think you deeply get it. The video barely shows superposition of waves and it only works in linear case.

  • @MarZandvliet
    @MarZandvliet 5 лет назад +6

    Impeccable timing! I'm working on various numerical schemes for ocean wave propagation for a videogame, and revisiting the very core physical principles at play is usually where the inspiration comes from. Thanks for bringing wave dispersion into focus, it's helping me out in think more clearly about difficult topics like error buildup and numerical dispersion artifacts.

  • @yordanmishev2764
    @yordanmishev2764 9 дней назад

    This is some of the most important videos ever created! thank you for your work, appreciate you all!!

  • @nathan6798
    @nathan6798 3 года назад +2

    WOW. Just wow. I dont think that this could have been explained more clearly. Thank you sooo much, this has helped tremendously! Keep up the good work and consider me subscribed

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  3 года назад +2

      Thanks for the compliments. I am glad my video was helpful and I am glad to have you as a subscriber.

  • @delphicdescant
    @delphicdescant 5 лет назад

    As always, you manage to describe these ideas with great clarity and precision. I much prefer your videos over videos of some guy waving his hands while trying to articulate these ideas in a "personal" way.

  • @basass101
    @basass101 5 лет назад

    The gold standard for explaining and visualizing math & physics since video #1.

  • @_fliszt4908
    @_fliszt4908 5 лет назад +1

    ありがとう
    こんな良い動画が見れる事がとても嬉しいです

  • @erictao8396
    @erictao8396 3 года назад +4

    This is a really helpful animation for a hard-to-visualize topic, thanks so much!

  • @adeshmishra7663
    @adeshmishra7663 4 года назад

    As usual, better than any other explanation on the whole internet.

  • @EugeneKhutoryansky
    @EugeneKhutoryansky  5 лет назад +11

    You can help translate this video by adding subtitles in other languages. To add a translation, click on the following link:
    ruclips.net/user/timedtext_video?v=EIqKG5TiSYs&ref=share
    You will then be able to add translations for all the subtitles. You will also be able to provide a translation for the title of the video. Please remember to hit the submit buttons for both the title and for the subtitles, as they are submitted separately.
    Details about adding translations is available at
    support.google.com/youtube/answer/6054623?hl=en
    Thanks.

    • @ka-md8ue
      @ka-md8ue 5 лет назад

      Hi, thanks for your videos! Do you need volunteer programmer for your videos?

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  5 лет назад +1

      k a, No. Thanks for the offer, but I make all the animations myself. I am glad you like my videos. Thanks.

    • @GiangNguyen-pe2dx
      @GiangNguyen-pe2dx 5 лет назад

      I need vietsub 😢😢😢

    • @tom_something
      @tom_something 5 лет назад +1

      Does the "group velocity" of component waves explain why light seems to slow down and photons seem to change direction when moving through glass?

    • @kseriousr
      @kseriousr 5 лет назад +1

      @@tom_something Yes. Original EM wave passing through electrons create a second wave. The resulting wave is slower than the original.

  • @aymanhassouna2981
    @aymanhassouna2981 5 лет назад +4

    OMG ...it is the first time to know that the standing wave is produced with this technique

  • @TheMostFacts
    @TheMostFacts 5 лет назад +1

    Thanks a lot! That is definitely an important video to understand this concept of wave velocity.
    I know you have millions of videos planned ahead already. But I just want to suggest that maybe you make more videos on solid-state physics, I think this is a huge part of physics that many students have problems with. I hope that in the future I'll see here videos about the Bloch wave, tight and weak binding, Brillouin zone, charge traps, more semiconductor physics and so on. That would be a dream come true!
    Thanks for what you're doing!

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  5 лет назад

      Thanks. Solid state physics is on my list of topics for future videos. I will add all the topics you mentioned to the list too. Thanks.

    • @morristgh
      @morristgh 3 года назад

      I know I'm commenting on a 2 year old comment but I study chemistry and my professor deduced the Bloch Theorem via Group Theory for us. Afaik it is usually taught different in the physics world and imo it is much simpler and more elegant via group theory. I also would appreciate more videos on solid state physics but I appreciate any video this channel uploads really.

  • @physicslover1950
    @physicslover1950 5 лет назад +11

    Well done sir u cleared a lot of my concepts. May u live long. Please now make a video on photo electric effect. Where we could visualize the photons coming with frequency same with the frequency of the valence electrons and then their collisions and see how photons of identical frequency push electrons from a metal surface by colliding with the electrons. And also dont miss to make a visualization of threshold frequency

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  5 лет назад +3

      Thanks. I will add the photo electric effect to my list of topics for future videos. Though, I briefly talk about the photoelectric effect in my video on nuclear physics at ruclips.net/video/r40h66qiF5I/видео.html

  • @martinpetersson4350
    @martinpetersson4350 5 лет назад +1

    Yes you finally made a new video! I love your stuff so much!

  • @therealDannyVasquez
    @therealDannyVasquez 5 лет назад +1

    Thank you, Eugene and Kira for your amazing work on these videos!

  • @yudhisthir7247
    @yudhisthir7247 5 лет назад +2

    Your videos always mesmerizes me. One more great video. Thankyou.

  • @nimraarshad4404
    @nimraarshad4404 3 года назад

    Best RUclips channel ... Love to learn from your videos.

  • @a.o.3523
    @a.o.3523 5 лет назад

    I always get excited when I see you post a new video! thanks!

  • @MrPranoybiswas
    @MrPranoybiswas 5 лет назад +1

    So nicely described..now all my doubts get cleared.. feeling lucky to subscribe this channel. Thank you Eugene and Happy New year.

  • @eigenchris
    @eigenchris 5 лет назад

    I never understood this in school. Thanks for the good visualizations.

  • @theunknown4834
    @theunknown4834 5 лет назад +15

    this gave me a WAVE of inspiration

    • @theunknown4834
      @theunknown4834 5 лет назад

      @Bob Trenwith well, my brain FUNCTION still works

  • @shareknowledgegainknowledg3557
    @shareknowledgegainknowledg3557 5 лет назад

    sir , no words , only standing ovation to u

  • @madagalaharivardhan9412
    @madagalaharivardhan9412 5 лет назад

    This channel is eligible for minimum 1MILLON views and subscribers

  • @StefanBurns
    @StefanBurns 3 года назад

    Very useful for better understanding passive-seismic physics

  • @mrfak8763
    @mrfak8763 4 года назад

    liked before watching just by looking at thumbnail
    edit: too good, just too good.
    thank you thank you thank you

  • @Mulkek
    @Mulkek 2 года назад +1

    Thanks, and it's so easy & simple!

  • @dumitruene8883
    @dumitruene8883 5 лет назад

    These animations are just beautiful.

  • @ambientsoda106
    @ambientsoda106 5 лет назад

    Exactly why I believe a mining tool can be created from waves...and why wave forms planes are a future general purpose product...literally your channel is what schools need desperately, classes can be augmented to allow for tablets and students can be free to explore, and now be tied to a class room

  • @marshallsweatherhiking1820
    @marshallsweatherhiking1820 2 года назад

    Very elegant explanation of how "wave-like" particles, such as those emitted by the sun during coronal mass ejections, travel at a speed 1000 times slower than the speed of light, yet have eerie, shimmery, wave-like behavior as they interact with the earth's magnetic field. There are few things readily visible in nature that travel at a speed in the middle of the continuum between the Newtonian realm and the relativistic realm, and also roughly in between quantum dual wave-particle realm, and the more strictly particle-like macro paradigm. Explains why auroras are very strange and complex.

  • @skun406
    @skun406 5 лет назад

    The behavior of waves can be counter-intuitive, your visualizations help to get a better grasp of the subject.

  • @aanandchaudhary5650
    @aanandchaudhary5650 4 года назад

    Wow now the imagination comes into existence with the help of these videos

  • @ronidaffan5904
    @ronidaffan5904 5 лет назад +1

    Eugene thank you so much for this video!
    This is an eye-opening animation!

  • @chuvzzz
    @chuvzzz 5 лет назад +1

    "when we have a wave describing a quantum particle, it is the group velocity that determines the velocity of the particle."
    This statement could use some further clarification, in particular in regards to the semantics of a particle's velocity.
    The total superposition is what is used in Born's rule, and this is the case whether we look at the position or momentum basis for the wavefunction. The wavefunction represents a wave of probability (e.g. the probability of a given configuration for a given particle), and in terms of the momentum basis, the group velocity refers to the location of the relevant envelope that covers the regions where the particle's momentum is most likely to be, even though the actual value of the momentum upon measurement may be any value given non-zero probability by the wavefunction.

  • @ernestng3
    @ernestng3 5 лет назад +1

    NIICEEE. Been waiting for this! Thank you!

  • @shirshak6738
    @shirshak6738 5 лет назад +12

    thanks eugene :D

  • @erikolsen1333
    @erikolsen1333 5 лет назад

    I love all your videos they got me through college

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  5 лет назад

      I am glad to hear that my videos have been useful. Thanks.

  • @ElektronikUygulamalar
    @ElektronikUygulamalar 5 лет назад +1

    Very nice explanation. Thank you very much.

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  5 лет назад

      Thanks. Glad you liked my explanation.

    • @ElektronikUygulamalar
      @ElektronikUygulamalar 5 лет назад

      @@EugeneKhutoryansky can you make a video about TEM, TE and TM wave propagation i guess this waves needs visualiton for better understanding.

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  5 лет назад

      This is not exactly what you are asking for, but I have a video on the polarization of electromagnetic waves at ruclips.net/video/8YkfEft4p-w/видео.html

  • @cubing7276
    @cubing7276 5 лет назад +8

    I never knew that there's a term for the speed of the regular sine waves and the speed of the resultant wave

  • @feynstein1004
    @feynstein1004 5 лет назад +1

    Excellent, as always. Can't wait for more :)

  • @efeplevneli9063
    @efeplevneli9063 5 лет назад

    Please never stop posting please

  • @rajns8643
    @rajns8643 4 года назад

    Very Very Amazing! God bless your soul.

  • @ErsanYolcu
    @ErsanYolcu 5 лет назад +11

    I wish I had this video back in 2009 when I was studying at uni! I would end up being a scientists.

  • @a_a_k_13
    @a_a_k_13 5 лет назад +3

    Love your videos.
    seen all

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  5 лет назад +4

      Thanks. I am glad to hear that you have watched all of my videos and that you like them. Thanks.

  • @connorleclair4011
    @connorleclair4011 5 лет назад

    best channel on youtube

  • @Trabzon3448
    @Trabzon3448 5 лет назад +1

    You were awesome, you are awesome, you will be awesome... Thanks :)

  • @lalitasharma6687
    @lalitasharma6687 Год назад +5

    So group velocity can be faster then light

  • @valor36az
    @valor36az 5 лет назад

    Extraordinary as usual

  • @seanwilliams7968
    @seanwilliams7968 4 года назад

    great video as always! Helps immensely with gaining intuition while studying.

  • @jeremiahmullikin
    @jeremiahmullikin 5 лет назад +4

    I really like your awesome animations. Can we please see a standing wave that results from waves of different amplitude and/or different wavelengths going in opposite directions?

  • @kustovas
    @kustovas 5 лет назад

    Спасибо Евгений, как всегда отличное видео!

  • @itsmidtrib1569
    @itsmidtrib1569 5 лет назад +45

    Who else has nothing to do with physics but watched these just for fun

  • @ShaolinMonkster
    @ShaolinMonkster 5 лет назад

    Good video for understanding before studying waves in physics

  • @silly_humor_memes2907
    @silly_humor_memes2907 3 года назад +1

    Such a great video. Thank you so much for making this video❤️❤️🔥

  • @rumfordc
    @rumfordc 5 лет назад

    highly effective educational material. thank you.

  • @primeobjective5469
    @primeobjective5469 5 лет назад

    Another work of art. Thanks!

  • @chuvzzz
    @chuvzzz 5 лет назад

    Nice! Go Eugene go! Love these

  • @ГришаГапон-д5ц
    @ГришаГапон-д5ц 5 лет назад

    ★★★★★(10/5)
    Great! Very nice job! Thanks! You are the better animator!

  • @zakirhussain-js9ku
    @zakirhussain-js9ku 2 года назад

    This phenomenon is cited as reason for slower speed of light in glass in a fermilab video. Can light induce secondary EM waves in glass and move at slower group velocity.

  • @NoosaHeads
    @NoosaHeads 5 лет назад +1

    Brilliant - absolutely brilliant.

  • @socratesuffer2765
    @socratesuffer2765 5 лет назад

    I wish there is a site that add waves on different amplitudes and phase velocities to enjoy the resultant wave randomness

  • @SorokinAU
    @SorokinAU 5 лет назад +1

    очень интересно и наглядно! большое спасибо!

  • @bn7809
    @bn7809 5 лет назад +2

    You made me love physics but now i am studying medicine coz it was too late to get into physics

    • @bn7809
      @bn7809 5 лет назад

      Uncle Garfield i was studying for my medical entrance exams. It was then when i came across these videos. I loved them, made me understand the depth of physics. But now since I am already in 3rd year of medicine and my country probably doesn’t work like that so idk if I can do that.

    • @michaelzimmermann3388
      @michaelzimmermann3388 5 лет назад

      Don't forget that physics includes a lot of mathmatics. Understanding the consequences (these videos) is only a small part of what you must do as a physicist. So before changing your life/study be sure you love math ;) and have some talent in it, cause otherwise wou will regret your decision for the rest of your life.

    • @MarZandvliet
      @MarZandvliet 5 лет назад +1

      It is never too late to get into physics. Do what you can, when you can. You may not be able to dedicate your life to it right now, or perhaps ever, but you'll be able to give it a place, and let it grow from there. :)

  • @gaurangagarwal3243
    @gaurangagarwal3243 5 лет назад

    Your work is amazing Eugene thank you for doing this. (:

  • @kid_flash97
    @kid_flash97 5 лет назад

    Thank you for this miracle!!!

  • @pranitrai9255
    @pranitrai9255 3 года назад

    Amazing explanation

  • @PETERLINNAH
    @PETERLINNAH 5 лет назад +5

    It seems that this would account for the occasional roque wave at sea.

    • @fizixx
      @fizixx 5 лет назад

      Exactly!

  • @mohitkwatra9587
    @mohitkwatra9587 4 года назад +2

    Can you explain in brief(somewhat intuitively) that why in case of electrons the shorter wavelength waves are faster but opposite in water waves.....?

  • @Numericthered
    @Numericthered 3 года назад

    Subconsciously ..As your back round music suggests an orchestra conductor is essentially the wave keeper. Whaling his hands about the air I guess isn’t purposeless. Seeing this video is when it dawned on me why he probably does this. As I may be wrong knowing nothing about an orchestra per say this video gave me an ahh hah moment. ..so. Different sounds aka groups of instruments must be kept in the same wave pattern to keep the flow moving smoothly. Since the person playing an instrument can only tune in on his particular wave sound. The visual affect of the conductors arms allow every one to know where the are in the wave. Keeping everyone in sync and making sure every one “goes with the flow”

  • @interstellarconveyance4865
    @interstellarconveyance4865 5 лет назад +1

    Thank you Dr. Khutoryansky!
    Because waves have the capability of carrying information, can complex packages of mathematical information be preserved in a wave form for very great distances? Or will the information degrade over the dispersion lifetime of the wave? How can a transmission into deep stellar space be preserved and transmitted without replicative fade or degradation? Certainly a carrier frequency of the magnitude required to send a package, would also disrupt or destroy the package by the very nature of its amplitude.
    Thank you Sir!
    Mr. Fractal

  • @schrodingersalphacat1862
    @schrodingersalphacat1862 4 года назад

    Thank you, thank you so much!!!

  • @chineduezihe4939
    @chineduezihe4939 3 года назад

    Wow.. Just amazing

  • @pranjalsharma3370
    @pranjalsharma3370 4 года назад

    Amazing! Thank You👍

  • @viniciusfernandes2303
    @viniciusfernandes2303 3 года назад

    Thanks for the video!

  • @shubhanshukaroliya
    @shubhanshukaroliya 5 лет назад

    Thankyou sir.... Again as always your video helped alot to build the concepts... Please make video on space time curvature.

  • @andreacuccia-k1c
    @andreacuccia-k1c Год назад

    please more video of waves… they are very difficult too imagine

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  Год назад

      I have many videos on waves. Some of them are listed below:
      Waves: Light, Sound, and the nature of Reality -- ruclips.net/video/Io-HXZTepH4/видео.html
      Diffraction interference patterns with phasor diagrams -- ruclips.net/video/NazBRcMDOOo/видео.html
      Polarization of Light -- ruclips.net/video/8YkfEft4p-w/видео.html
      Quantum Wave Function Visualization -- ruclips.net/video/KKr91v7yLcM/видео.html
      Phase Velocity versus Group Velocity -- ruclips.net/video/EIqKG5TiSYs/видео.html
      Art made with wave interference -- ruclips.net/video/_7XNu2lUKIA/видео.html
      Sonic Boom -- ruclips.net/video/lGlyNl1srAc/видео.html
      Electromagnetic Waves -- ruclips.net/video/W1cTpqM9DaU/видео.html
      Beat Frequency -- ruclips.net/video/lH6hRdmY0DY/видео.html

  • @sayan_debnath
    @sayan_debnath 4 года назад

    Thanks, it was helpful 🌹

  • @fredflintstone8003
    @fredflintstone8003 5 лет назад

    Great visualization

  • @fractalnomics
    @fractalnomics 3 года назад

    Perfect. I can use this.

  • @pasindumalinga1098
    @pasindumalinga1098 5 лет назад +1

    Very useful

  • @sebbes333
    @sebbes333 5 лет назад +2

    Is there some kind of algorithm or something, that can de-construct the "resulting wave" into its many "component waves"?

    • @emiliosereni1864
      @emiliosereni1864 5 лет назад

      Nice question. I'll wait for an answer too

    • @koharaisevo3666
      @koharaisevo3666 5 лет назад +8

      Like Fourier transform?

    • @alexburgess2515
      @alexburgess2515 5 лет назад +2

      Watch 3Blue1Brown's videos on Fourier Transform if you want some high quality animation, he does a great job explaining that concept

    • @musaire
      @musaire 5 лет назад +1

      Yeah, I am sure Fourier had a solution for this. It is not too complicated, you can roughly even start matching waves to the group wave by your eye, starting from the biggest wave you can fit in it, then scrap the rest and fit smaller ones in it, etc.

    • @AdityaRaj_24
      @AdityaRaj_24 5 лет назад

      Yes that algorithm is Fourier transform. You put in a given wavefrom and it gives the amplitude of different component plane waves of each possible wavelengths.

  • @Toocrash
    @Toocrash 4 года назад

    We want this we need this. ty

  • @maunil108
    @maunil108 5 лет назад

    Superb video

  • @robertcalkjr.8325
    @robertcalkjr.8325 5 лет назад

    Very nice! Thanks!

  • @Upendra3737
    @Upendra3737 5 лет назад

    Nice explanation

  • @nvkeerthana760
    @nvkeerthana760 2 года назад

    Why is phase velocity greater than the speed of light and yet does not violate any principles ?

  • @jingcineo5815
    @jingcineo5815 4 года назад

    Why are the shorter wavelengths faster for quantum particles and slower for water waves?

  • @aymanhassouna2981
    @aymanhassouna2981 5 лет назад

    really .i love this channel soooooo much ur content is very important & unique ..
    truly..i wanna help to share & spread science .i can help to translate this div into Arabic.....i'm Egyptian..

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  5 лет назад

      Thanks. You can help add Arabic subtitles for this video by clicking on the following link ruclips.net/user/timedtext_video?v=EIqKG5TiSYs&ref=share

  • @hannah-xo5ss
    @hannah-xo5ss 5 лет назад

    awesome video! and only a week too late after i bombed my physics final

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  5 лет назад

      Glad you liked my video. Sorry I didn't make it one week earlier.

  • @jeffryblackmon4846
    @jeffryblackmon4846 5 лет назад

    Thank you.

  • @SuperStingray
    @SuperStingray 3 года назад

    It's like how a "traffic jam" propagates backwards while cars drive forward.

  • @godther1391
    @godther1391 6 месяцев назад

    you are the goats

  • @shanaya5855
    @shanaya5855 5 лет назад

    Very nice

  • @ganeshbandgar881
    @ganeshbandgar881 5 лет назад

    Thank you 🙏🙏

    • @EugeneKhutoryansky
      @EugeneKhutoryansky  5 лет назад

      You are welcome and thanks.

    • @ganeshbandgar881
      @ganeshbandgar881 5 лет назад

      @@EugeneKhutoryansky feeling happy because your reply ...
      I want to say something logical but feels like illogical .
      1) intelligence can get frustrate with questions...?
      2) Using Our intellect , Can we know what we want to know ?
      3) what is the propose of all this things - thinking , Group velocity , Maxwell equations, air , water, ........ endless list. ( Ans : to break our boundary ?)
      Thank you .
      I am not here to prove I am intelligent.
      I saw all your videos , one of the video shook my intlect at its peak. And i get doubt full all time.
      After lots of time ( months, years)
      I came to know about the one person , he put me in the direction which is .....( End thinking capabilities).
      Thank you .

  • @petruspetrusuom3342
    @petruspetrusuom3342 5 лет назад

    thank you

  • @KBtek
    @KBtek 5 лет назад +6

    Hi Good evening from India

    • @p.g.pg38
      @p.g.pg38 5 лет назад +2

      And good afternoon from France 😜

    • @p.g.pg38
      @p.g.pg38 5 лет назад +1

      @Robert Rodgers the earth is therefore not flat? 😂

    • @erikolsen1333
      @erikolsen1333 5 лет назад +1

      Good morning from Asheville North Carolina USA

    • @erikolsen1333
      @erikolsen1333 5 лет назад +1

      @Robert Rodgers been there hell yea have family in marion

    • @mostafakamal753
      @mostafakamal753 5 лет назад +1

      Good afternoon from Egypt