8.03 - Lect 11 - Fourier Analysis, Time Evolution of Pulses on Strings

[@bobwhite137]

What a wonderful introduction to the world of Fourier analysis! For those wishing to really wrap their heads around the far reaching utility of this type of analysis, pick up a good book on linear systems. You'll see the beautiful relationship of Fourier and Laplace transforms, and their discrete versions (like the Fourier series in this lecture) and z-transforms as the discrete analog of Laplace transform. Their use in signal analysis, information theory, control theory, communications systems gives absolutely amazing insights. Thinking in the frequency domain, with orthogonal/orthonormal components has yielded some of the most amazing practical inventions of the past 40 to 50 years. A real testament to Fourier and Laplace that only recently have we figured out how to truly capture the beauty of their theory. Many thanks for this lecture.

[@ShubhamBhushanCC]

For long time I have been tired of physics, considering I got a graduate degree. Today after 6 years, I feel old passions stir in my heart. I am so thankful to you Prof. Lewin, thanks to you I am singing Pina colada song for physics.

[@AlpoDers]

You made me understood Fourier and even made me interested about it. Thank you so much for your lecture, especially the demonstrations which helped out tons. I truly believe that you make a difference in physics for a lot of different people, changing their viewpoint. Honestly, this video was quite refreshing.

[@lecturesbywalterlewin.they9259]

Glad to hear that.

[@andrepadilha5454]

Hello Mr. Lewin. At 58:45, probably the reason that people "hates" the 7th harmonic is because it forms, with the fundamental tone, a tritone in music. A tritone is an ambiguous interval, and in the study of Classical Music, a tritone ALWAYS has to be resolved. A looong time ago, tritone was even called "the Devil interval xD", the musicians used to listen to a tritone as an unpleasant sound. However, in modern music (like Blues and Jazz), the 7th is well-accepted. Hugs from Brazil!!!! PS: I'm doing my Math Master's research at Fourier Analysis! Loved this lecture!

[@신현재-u9z]

It is excellent to understand the nature with your lecture.Normally we learn the physics for testing so we do not know the essence of concept of nature. Thank you for your lecture to open to us in global. God bless you. 😀

[@lecturesbywalterlewin.they9259]

It's my pleasure

[@jamesr3505]

Having watched all of the 8.02 lectures and the 8.03 lectures up to this point, I am amazed at the beauty and simplicity of the physics perspective. Only here does it become very painful not to do things like a mathematician.

[@jamesr3505]

In particular, the explanation that the square bump cannot have any terms with cosine is simply wrong. It requires global symmetry (evenness vs. oddness, not just looking at the behavior near 0).

[@mannyquinn9031]

@@jamesr3505 bruhh, evenness and oddness of functions can be told from the geometric origin (reflections about the x and y axis around the zero point). and for his "periodic" example he is totally right using the origin as his only reference point because we can assume that the function is "globally" repetitive since it is "periodic" hence one period is enough to make a solid conclusion

[@realcygnus]

Amazing that Fourier Analysis was conceived of before "electronics"........so many practical applications ....many of which would most likely be impossible otherwise. .....its near the top of my list of useful maths/technologies !

[@lecturesbywalterlewin.they9259]

:)

[@justlearn149]

Fourier was solving heat transfer problems using his series

[@ianmichael5768]

@@justlearn149 Sad really. Energy lost. Never to be used again...

[@ryanoberoi454]

Thank you sir! you are like a teacher I never had..this video really helped me to understand Fourier analysis.

[@lecturesbywalterlewin.they9259]

Ryan Oberoi Thank you! Statement like this are very rewarding

[@rohitraj4275]

I love Fourier analysis, I love this human for making me fall in love with all these things. I got a great research opportunity as a freshman, all because of him. Otherwise I would have been sitting in remote village of India where even Internet connection is not stable.

[@harshraj402]

you sir ,are the greatest gift bestowed on to us : The Students LONG LIVE PROFESSOR LEWIN!!

[@Miguel-tr2ev]

Watching 18.03 lectures was extremely helpful understanding this lectures. But nothing beats seeing it being applied in physics.

[@juanmanuelferrero3017]

THANK YOU VERY MUCH!!! I realy enjoyed this lecture. Hope to meet you. I consider you the best teacher that a pupil can have. I'll do my best to spread your love (and of course mine) for Physics

[@TheSaptek]

thank you sir! now half range series of Fourier analysis is quite clear!

[@robertoortega1801]

This is really a Magister class... Thank you so much. I wish I would have access to this first class material when I was a student.

[@call_me_stan5887]

How come this lecture has only 16 thumb ups? Fourier analysis and Fourier transform are one of the most underestimated tools in modern physics... because they became so ubiquituous! It's amazing what one can achieve using these tools. And of course one of the most important application is in signal analyzers and signal processors. In fact, signals like this one: i.imgur.com/uD0wyAT.gifv would not only be impossible to analyze, but also impossible to produce, without a thorough Fourier analysis involved (notice steep filters and also notice the signal is clipped by multiple dB's at the very same time). To me it's a masterpiece, since you can't see any spurious harmonics in the spectrum (even at -90 dB) but just baseband part, pure 19 kHz pilot tone and finally the sidebands that contain the difference of L-R part of the input stereo audio and RDS (because it is in fact a multiplex signal coming out of the audio processor with integrated stereo coder for FM radio and looks exactly like in the book, if not better, because with such steep filters you could go all the way up to 16 kHz in audio bandwidth - pretty much upper limit of frequency response for FM radio: upload.wikimedia.org/wikipedia/commons/thumb/c/cd/RDS_vs_DirectBand_FM-spectrum2.svg/744px-RDS_vs_DirectBand_FM-spectrum2.svg.png).

[@souvikpaul3312]

Thank you very much sir .... probably I was trying to understand the physical meaning of Fourier series,why and from where it comes but everywhere in KZbin they are just talking about some formulas of Fourier series 😥..... this video helped me a lot😄

[@carlos43g]

It´s quite rewarding to watch these videos. Personally I´m reviewing these subjects and no one can beat Lewin to make it so clear !!! I remember two fine books that helped me a lot with Fourier series and integrals, Hwei hsu and Churchill .You and Mr Feynman are my inspiration!! have you ever met him mr Lewin ? regards

[@lecturesbywalterlewin.they9259]

I am the REAL Walter Lewin. I run this channel.

[@carlos43g]

Nice to know you are the real one !! i meant if you ever met Mr Feynman :-)

[@CatsBirds2010]

Best Fourier analysis ever!

[@somyaswarnkar8623]

This is a very helpful lecture sir. I was helping a mate in Fourier analysis and she asked me, as you said at 47:45 that at this moment the string has zero pulse, than why does anything happens further? the string is flat. No pulses! what's the motivation?

[@hannaedwards5197]

Around 39 minutes it struck my mind that no matter how many terms you add , at the locations x=0 and x=L , the value of the function will always be zero , because the value of all its components is zero at these points . So this wave is physically impossible . Not ? Or mathematically - our function if we sum up an infinite amount of terms to "at infinity" (equivalent to introducing that point on top of Riemann's sphere) - this function would then be an "exact" straight line at +a and -a but have the value zero at 0,L,2L... . Right ? .. Looking forward to enjoy more of this video later ! So important to restudy things from different sources . This really helped me more .

[@lecturesbywalterlewin.they9259]

>>> no matter how many terms you add , at the locations x=0 and x=L ,>>> of course But the more terms you add the better will ne your fit to the square function.

[@tehyonglip9203]

thank you very much for these wonderful explanation, fourier analysis is so important in all fields of physics, especially quantum mechanics.

[@lecturesbywalterlewin.they9259]

:)

[@surendrakverma555]

Excellent lecture Sir. Thanks and Regards 🙏🙏🙏🙏🙏🙏🙏🙏

[@akashsunil7464]

Hey dr walter lewin in the a^2+b^2where you plotted square of amplitudes vs omega i understand what u said about how they detected the pulsars but how did the original graph look like as a function of x and moreover how did the scientists know that the pulsars spun at that particular frequency omega

[@heiheihei60]

Good to know the huge application of Fourier analysis

[@LongNguyen-ou7br]

Dear Dr. Walter Lewin, Thank you for the amazing lecture! I want to ask you how you can simulate the Fourier waves at 51:16. I really want to demonstrate for myself. Thank you so much!

[@lecturesbywalterlewin.they9259]

it was a program written by one of my graduate students.

[@LongNguyen-ou7br]

@@lecturesbywalterlewin.they9259 Thank you for replying, Dr. Lewin. Do you still have the code for it that you could share to all of us?

[@sagarrathore5400]

@Long Nguyen Any luck finding or constructing that code yet?

[@sanjay3039]

Harmonics are beautiful!

[@sebasojos]

Great Thanks!. BTW, in the video 53.47 is the real square shape we are analyzing.

[@sebasojos]

not the one in 54.08

[@ElectromecanicaIndustrial]

Amazing Lecture , wow!

[@lecturesbywalterlewin.they9259]

Glad you think so!

[@jeffgiebens3197]

U bent een zeer interessante persoon !

[@BinhNguyen-if5yd]

In reality where we can see something like things at 51:31: We let a pulse on a string go and then we see two pulses with half amplitude propagating in the opposite directions? I think i've never seen it before.

[@physl2787]

Sir please send mit notes on fourier serie

[@konjaleo]

Where is it possible to find this great code? I am a physics teacher and it would be fantastic to show this to my students!

[@lecturesbywalterlewin.they9259]

question unlcear

[@konjaleo]

@@lecturesbywalterlewin.they9259 where could I find the simulation that you are showing at 51:40?

[@konjaleo]

Is it avaible online somewhere?

[@lecturesbywalterlewin.they9259]

contact [e-mail] refer to my lecture and how many minutes into the lecture.

[@kabirsingh8176]

From 18:21 to 18:51, why is f(x) not replaced by f(πx/L) ?

[@shreyashmeshram5351]

do we use calculus to derive such a large general Fourier transform???

[@manuferre7186]

Really inspiring. Thanks Professor!!! ;)

[@lecturesbywalterlewin.they9259]

My pleasure!

[@Irfan-vo6fh]

Where can i get that software you used in lecture

[@emirselman3984]

I dont get how term 2 and 3 become zero If you fx take the integral of A_mcos(mx), why do you get 0 instead of A/m*sin(mx)+C?

[@TonalWorks]

Not "instead of". Your primitive function is correct. But we integrate it from -pi to pi. It's a definite integral. The sine of pi and -pi are both zero.

[@mnia8917]

very good. Thanks a lot.

[@tushardey7653]

Fabulous sir

[@dahmouniproduction9804]

thank you so much sir it's really helped a lot

[@lecturesbywalterlewin.they9259]

:)

[@Cirius9]

36:45 Me and my classmate during the Lecture

[@Shoaib14972]

thank you very much sir!

[@physl2787]

What is difference between harmonic and normal mode

[@lecturesbywalterlewin.they9259]

no difference

[@mgshweyoeshweyoe4483]

Thank Sir , I got It.

[@kaustavtamuly696]

39:00 I kinda felt some goosebumps..!!

[@sytk3694]

Thanks to you sir!

[@muzzammilb3437]

Thankyou sir!

[@lecturesbywalterlewin.they9259]

Most welcome!

[@murnow]

love u love u love u u are amazing

[@lecturesbywalterlewin.they9259]

:)

[@hannaedwards5197]

Just after 50 minutes : but what would happen if we let go of the square but it is fixed at its ends ?

[@lecturesbywalterlewin.they9259]

To mimic "properly" the square with Fourier components we need thousands of frequencies. Ideally an infinite number of terms. We can then change y in zero time at x=0 and at x=L. Each term is then a standing wave and we can see its progression in time. If we wait long enough, the square will have flipped over (just like the triangle). I watched my demo of the "traveling" triangle! FABULOUS only 45 terms! A triangle also changes y in zero time at 2 places.

[@hannaedwards5197]

Lectures by Walter Lewin. They will make you ♥ Physics. I meant what if the square makes up the entire string alike your mathematical sample, and as opposed to your animation?

[@lecturesbywalterlewin.they9259]

I can never squeeze a string into a square as that would require an infinitely large force at x=0 and at x=L. I have no choice, I must allow some delta x to go from y=0 to y=y_max at x=O and to go back from y_max to zero at x=L. I can ONLY make delta x = 0 at x=0 and at x=L in a mathematical exercise.

[@hannaedwards5197]

Lectures by Walter Lewin. They will make you ♥ Physics. Yes this is true that you cannot squeeze the string into a perfect square (this relates to my other comment), but, you need not leave a nonzero delta x, but merely the sides would not be exactly vertical. ... Because even with leaving a delta x, you still cannot make the sites vertical, can you, for your own reasoning.(?)

[@hannaedwards5197]

And in my comment below, in my mathematical exercise, it also doesn't work, because "at infinity" we get a function that is f={ a if 0

[@jaysethi733]

I'm skipping this topic nothing personal tho

[@adamfattal468]

Jay Sethi Out of any topic, you choose this?

[@mwaqar2182]

❤