Deriving the Wave Equation

[@CallOFDutyMVP666]

The fact that I can access this high quality of a lecture for free is astonishing..

[@afammadudaniel2982]

I'm practically blown away!! I'm at the phase of research where I'm trying to understand how PDEs are embedded in Machine learning loss function. Viola! Here I am consuming mathematical chocolates!

[@anishsharma6702]

same , i love this kind of detail approach to concepts , with implementation of human logic and human intuition at grond level.

[@dennislui2938]

Agreed. I always thought Gilbert Strang at MIT is a great math teacher. Steve has proven to be just as good, if not better, than Prof Strang. Kudos to eigensteve 🙏🙏🙏

[@paulosimones3]

Thanks Steve. For many years I have dealing in higher maths subjects and, honestly, this is one of the best lectures I have watched (or physically attended). Please, keep producing tuition material at this level of excellency.

[@rajinfootonchuriquen]

From a geometrical standpoint, the laplace equation means: "scalar field without local max and min"; heat equation means: "the change in one variable is proportional to the curvature in another"; and the wave equation means: "the curvature in one variable is proportional to the curvature in another". If you can imagine how the information change, you can easiliy derivide this partial differential equation.

[@charlesschmidt4272]

I had the exact same block when starting to learn about PDEs. This derivation is so crystal clear about what assumptions are being made and why they are made.

[@alilabbene8166]

Very intuitive and easy to understand. I also appreciated the emphasis on how it was not that easy to finally get comfortable with the manipulation of such an equation. Thank you very much for this video and for the whole channel 🙏

[@clairezhao8963]

This video series explains why it's harder and harder to resist binging KZbin these days, any other series like this? The new videos are literally in sync with my PDE class oh my goat

[@timy8749]

I do envy the new generation who can study PDE with high quality reference like this. It took me years to think through some of the concepts.

[@khanster]

I'm taking PDE this semester and your PDE playlist has been awesome. Thanks prof.

[@hajsh67]

Awesome intuitive approach to setting up the wave equation from F = ma. Reminds me of my General Physics course when I was reading the Young and Freedman text.

[@manirarebajeanpaul9312]

Thank you so much Steve, its like reading a very huge book in a short moment.

[@TNTsundar]

This is next level lecture. Love your videos. 👏

[@Pier-zl7gm]

Very neat introduction to the wave equation, well done prof ! One could add - just for more fun - that those smart mathematicians from the 18th century wrote all their ‘papers’ in Latin and so the obvious symbol choice to represent speed was the letter c .. speed being “celeritas” in Latin. Funny how Latin even got into the most famous among all equations, even if Albert Einstein didn’t use c initially 😊

[@liorcohen5833]

What I didn't get the first time I saw this derivation is why the length of the rope is dx (in the context of the mass). it's actually: sqrt(dx² + dy²) = dx•sqrt(1 + (dy/dx)²) = dx•sqrt(1+(y')²) But since we assume small oscillations all nonlinear terms are negligible so ds = dx•sqrt. It's similar to us saying cos(θ) = 1 and not 1 + θ²/2! + .... Hope this helps someone!

[@miroslavvorechovsky1370]

A very nice lecture, thank you! I have a minor comment though regarding the derivation. When considering the force equilibrium of the infinitesimal element, I am afraid the equilibrium in not maintained if the two tangential forces T at the ends are identical. What must be identical to prevent horizontal movement are the horizontal projections of these forces, say “N”. When these horizontal forces are identical, the vertical projections of the tensile forces T, which we can call F, are equal N*tan(theta). And it is the difference between these vertical forces: N*[tan(theta+dtheta)-tan(theta)], which equals the Newton’s inertia forces “m*a”. This is just a minor fix which removes the weak arguments (time 20:20) about sine being roughly equal to tan, which is equal to the angle itself, and cosine being roughly equal to one for small angles theta.

[@danialheidar8847]

I don't know how to say thank you to making my nightmare to day dream, Wish i was your student and learn this things directly in your class

[@murillonetoo]

Great explanation, professor! I'm looking forward to see the upcoming videos!

[@SuyueYuan]

what a beautiful lecture!

[@BioMedUSA]

Played the intro a couple of times - Nice segue!

[@kepe7323]

Fantastic intro

[@sam78ize]

these lectures will keep on giving into the future. you are doing a great service. some professors should also take your classes. 😂

[@okhan5087]

Thank you for this playlist. Your videos are helping me a lot in my PDE class.

[@shsaa2338]

How we derive that c^2 = T / ro? In this video Steve is explaining how to derive the wave equation Utt=c^2 * Uxx - correct? From F=ma Steve derives Utt = T / ro * Uxx, and then at 25:15 he just says “where c squared is equal to T (tension) divided by ro(linear density)”. Where that comes from? How speed comes into the equation?

[@camwhite113]

Thanks for making this content openly available! It has certainly been extremely helpful while brushing up my memory on these concepts. I did have a question: Couldn't we skip the sin(theta) ~ tan(theta) step altogether by utilizing the requirement that the x-component of the tension at points x and x+dx must be equal (in opposite directions)? At either point, we have tan(theta) = T_y/T_x. Solving for T_y, we have T_y = T_x*tan(theta). Again, T_x is the same at both x and x+dx (save for the minus sign), so it can be factored out when calculating the net vertical force, F = T_y(x + dx) + T_y(x) = T_x*[tan(theta + dtheta) - tan(theta)]. Thanks again!

[@awsomeguy3291]

First, second, third times watching this: _crickets_ Fourth time watching this: "YOOO THAT DERIVATION IS SO COOL!"

[@rushabhyeshwante]

I did find it interesting and fun. Thank you for simplifying the concept.

[@Joeleo]

These are awesome Steve, great work.

[@spsorn5433]

Thank you very much for your fantastic lecture and your hard work. I love it.

[@muhammadkahshan6216]

Excellent explanation. Crystal clear. Thank you

[@anuchita6979]

Very good teaching. From thai student

[@saadhassan9469]

Beautiful Lecture and Wonderful Lecture series!

[@pain4743]

Amazing, Thank you

[@marsgao4084]

OMG the guitar is so cool!!

[@Eigensteve]

Thanks!! So much fun to play

[@ananthakrishnank3208]

As for now, I am not convinced on neglecting cosine components of T, saying theta is close to zero. Just as I type, maybe I get it. For angles x1 = 0.01 to x2 = 0.02 (tending to x1), we are kind of looking for sin(x2) - sin(x1). From differentiation we know that sin(x2) - sin(x1) = dx * cos (x1). So analogously for cosine components, cos(x2) - cos(x1) = dx * sin(x1) (approximately zero for very small x). So, this way, it makes sense to only include the sine components. But now why this is not convincing is, for x = 10.00 to x = 10.01, I cannot justify the neglection of cosine component. Ooh. I get it. First of all, when we visualize the movement of a guitar string, we see that it's as if the string just moved a tiny bit up, even when its up it looks still flat. In that case, theta is obviously close to zero. So I should have not even considered x >> 0 case. Thanks for the lecture! :)

[@greenfoodpower6961]

Excellent explaination! Correct pace.

[@kelvinadimaswijaya9523]

wow this is walter lewin's level of lecture, thankyou sir

[@Tom-sp3gy]

You are the best ever !

[@loadingUserID...]

Excellent and practical video on the topic.

[@AleeEnt863]

Love you, Steve! Blessed!

[@belatar]

also i hope you plan to do this all the way to the schrödinger equation :)

[@klammer75]

That would be epic! Please do sir!🤩🥳🤓

[@Eigensteve]

I’m working up to it. Might take a little while. Navier-Stokes equations will be sooner.

[@belatar]

@@Eigensteve 😍😍

[@barakgavriel1028]

Amazing video! Thank you 🙏🏻

[@lazarbaruch]

You are right that model building is practically inexistent in both mathematics and Physics curricula. Do yourself a favor and read G. Strang's "Applied Mathematics" and Enzio Tonti's papers. Read all the stuff about across and through variables and how this equation is just the continuous version of a discrete electric circuit. Then, students will also get a better understanding of the meaning of curl and div operators.

[@malikialgeriankabyleswag4200]

You are a legend

[@micahdelaurentis6551]

why can you ignore the very small vertical distance when you say rho*delta x = mass, but you can't ignore the very small vertical distance when you assign non-zero angles?

[@ramazansubas1684]

I am looking for a video on how to generate the Energy Functional of a partial diff equation. How to generate the energy function of a wave equation. Is this video about that?

[@Tyokok]

You are not only a great professor, but also a great person!!! Great great Thanks and God Bless!!! One quick Q: why that "c" is the speed of wave? any derivation reference? Thank you!

[@sorry4all]

maybe c for constant?

[@Tyokok]

@@sorry4all thanks for reply. not sure if c is constant. no what it is or not, what's the physics meaning of it?

[@ananthakrishnank3208]

@@Tyokok Dimensional analysis. Tension is just force (Kg .m .s^-2). Linear density (Kg .m^-1). Now when you divide both, the units you are left with is that of speed squared. Clearly the constant is indicative of some speed. I can only see two kinds of it, one along the medium, another is that of string's up-down vibration which decays in the process. But speed of wave stays constant till the end, just like on whipping the long rope, the bump moves with same speed till end. Anyone can correct me.

[@philoso377]

Nice video and presentation. Are we talking about mechanical wave or not?

[@ernstuzhansky]

Thank you!

[@quantum_mechanics253]

love it!

[@JoaoLima-pq1hm]

Happy 2024, thank you so much for this excellent lecture. 🎉

[@sorry4all]

Woah now that was really satisfying.

[@andyowen3685]

I expect a full concert in the next video

[@Eigensteve]

Haha yeah… I’m not going to quit my day job…

[@thomasjefferson6225]

I'm loving this pde kind of mathematics Im sorry, but im not thinking any of this is easy man. This stuff was made by great minds.

[@favicon2978]

absolutely true

[@hamsterpoop]

I have a series of videos on my channel about deriving the wave equation from Walter Lewin at MIT

[@seanwrfps]

Thank you, this was helpful!

[@jcsjcs2]

I think going the step to set sin(...) = tan(...) is unnecessary. You have the force T going to the left at and the same force going to the right at the other end of the segment of the string. You can assume that the "horizontal" component of the force is equal at both ends. Otherwise, the hole segment of string would start to move sideways. Set that force equal to "T" (because of small angles and cos being equal to 1) and the need to argue that sin=tan will disappear. To me that seems a bit tidier and easier to follow.

[@sorry4all]

I think it's necessary. tan allows you to change the variable from θ to x. And to be more precise (rather than to just call it an approximation,) In the limit, sin(θ) = sin(θ)*(tan(θ)/θ) = (sin(θ)/θ)*tanθ = tanθ Since Lim θ-> 0, sin(θ)/θ = 1 Lim θ-> 0, tan(θ)/θ = 1

[@fahrenheit2101]

i don't see the difference. in either case, you heavily lean on small angles

[@jcsjcs2]

@@fahrenheit2101 To me that seems a bit tidier and easier to follow. But it assumes that there is no sideways motion of the particles.

[@ElizabethMuringi]

❤ thanks. God bless you ❤️.

[@JonatanEngstrom-kd7it]

crazy good

[@damiangames1204]

High quality

[@GimbertLane]

I don’t think you ended up explaining why this is considered a hyperbolic differential equation. I would love to understand what types of differential equations are elliptic, parabolic and hyperbolic.

[@Eigensteve]

Good point. I will be making a video on this soon I hope.

[@klammer75]

Bravo👏🏼🎓

[@Eigensteve]

Thank you!!

[@jakubsebek]

Beautiful!

[@robertparrish9014]

Thank you so much!

[@USFJUM]

😂I love you.

[@Joseph-Lau]

That’s an interesting lecture that makes me to revisit my knowledge of physics again. The tangent of angle is equal to the Uxx (x,t) is kind of tricky which I need to do some revision. Do you have any idea about the speed of the wave if it is a probability wave in quantum physics?

[@kilianklaiber6367]

If you want to derive the net force on a small segment of the string, then you have to add all of the forces acting on each infinitesimal part, correct? But instead you subtract two forces from each other. This must be the result of the integration. Therefore, I believe that something is missing in this derivation.

[@johnwesley4713]

That is exactly what he’s doing. He defines a differential unit, x to x+dx, and he assumes that the only forces acting on this section are the tensions of the string of the unit cell either side of this section…then he sums those forces, which since they are acting in opposite directions becomes a difference.

[@ajstube54]

super clever :)

[@jonludwig8233]

You mentioned the wave being infinitesimal and the effect on entropy, any clues where I can follow up on that idea?

[@leepatrick1756]

Thanks! I get it!

[@belatar]

really great video but currently i struggle at one point: where you let dx ->0 to me, that would just let the term 1/dx grow infinitely large but instead you „define“ this as Uxx and consider this clear. unfortunately i cant follow that step, so could you please explain that step in some more detail?

[@press2701]

It's a logic point. As dx->0, so does du->0, in the limit du/dx is finite. Go back to fundamental theory of calculus (FTC) for the complete story (which is tedious).

[@Eigensteve]

The answer below is the right idea. We are essentially using the definition of a derivative, which has some assumptions involved.

[@belatar]

@@Eigensteve damn, 20 years ago i would have remembered 😭 its all too long ago, but thanks for responding.

[@rjaph842]

Hi, Steve. Thanks fr the great video. A quiz here, isn't X a function of time?

[@shsaa2338]

X could be expressed as function of t for each particular U. Or more generally X is a function of two variables - t and U. Because that equation has 3 variable - You could express any variable as function of two others.

[@muthukamalan.m6316]

❤️❤️

[@AimtAprilMyo]

What is deflection

[@leepatrick1756]

Hi. Can someone explain how the limit becomes the second derivative. I would reallt appreciate this, as then i would fully understand. Lee

[@bramilan]

Look for the definition of a derivative - it's exactly what we have here.

[@tolkienfan1972]

LC oscillators in electric circuits too. This is not hyperbole!

[@estebanmeneses3107]

Hello, first thank you so much for these amazing videos. I am very rusty at math, but could somebody tell me why second order linear ODEs have exponential or sines solutions? I would really appreciate it.

[@dennisgawera8788]

Because derivatives of sines and exponents to the base e are also sines and exponents to the base e of the same order. This makes the solutions just combinations of the same sines and exponentials.

[@estebanmeneses3107]

@@dennisgawera8788 Thank you so much!

[@حسینکیوانی-ز5و]

Is it true for big deflections?

[@kevinni7214]

i love you

[@danielvolinski8319]

What part of this explanation you did not understand when you were a teenager?

[@Eigensteve]

It just felt very dry and unmotivated. I don’t think I intuitively understood the assumptions and I struggled with the partial derivatives and what they meant physically.

[@ravenecho2410]

@11:30 me too, i have much better math skills (and was able to do the cosine thing now -insulated boundaries, but idk always found PDE hard as self study😢) fingers crossed!

[@favicon2978]

🥲

[@vansf3433]

nothing new. it looks like for high school levels

[@lksingh8122]

You still need to learn more mathematical physics, some of your fundamentals are still not clear to you 😢

[@AimtAprilMyo]

What is deflection

[@SuyueYuan]

I think it means the maximum deviation of the string from its rest position. Basically the highest point of that string you see in the plot.