Deriving the Wave Equation
Вставка
- Опубліковано 24 чер 2024
- In this video I derive the Wave Equation, one of the most important and powerful partial differential equations. It can be used for a huge variety of other problems in physics and engineering. In the next few lectures, we will solve and analyze this equation. Examples from a guitar string are used to illustrate ideas.
@eigensteve on Twitter
eigensteve.com
databookuw.com
This video was produced at the University of Washington
%%% CHAPTERS %%%
0:00 Overview
2:00 The Wave Equation and Examples
8:33 History of the Wave Equation
10:03 Deriving the Wave Equation from F=ma
25:25 Quick Recap of Derivation
31:53 The Wave Equation and the Guitar String
35:08 Conclusions and Next Videos - Наука та технологія
The fact that I can access this high quality of a lecture for free is astonishing..
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!
same , i love this kind of detail approach to concepts , with implementation of human logic and human intuition at grond level.
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 🙏🙏🙏
Great explanation, professor! I'm looking forward to see the upcoming videos!
These are awesome Steve, great work.
This is next level lecture. Love your videos. 👏
I'm taking PDE this semester and your PDE playlist has been awesome. Thanks prof.
This video series explains why it's harder and harder to resist binging UA-cam these days, any other series like this? The new videos are literally in sync with my PDE class oh my goat
Thank you so much Steve, its like reading a very huge book in a short moment.
Excellent explanation. Crystal clear. Thank you
Thank you very much for your fantastic lecture and your hard work. I love it.
I did find it interesting and fun. Thank you for simplifying the concept.
Thank you for this playlist. Your videos are helping me a lot in my PDE class.
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.
Beautiful Lecture and Wonderful Lecture series!
Excellent explaination! Correct pace.
Excellent and practical video on the topic.
Love you, Steve!
Blessed!
Amazing video! Thank you 🙏🏻
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
Fantastic intro
Thank you, this was helpful!
Woah now that was really satisfying.
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?
these lectures will keep on giving into the future. you are doing a great service.
some professors should also take your classes. 😂
Amazing, Thank you
Beautiful!
You are the best ever !
Thank you so much!
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.
love it!
Thank you!
wow this is walter lewin's level of lecture, thankyou sir
You mentioned the wave being infinitesimal and the effect on entropy, any clues where I can follow up on that idea?
also i hope you plan to do this all the way to the schrödinger equation :)
That would be epic! Please do sir!🤩🥳🤓
I’m working up to it. Might take a little while. Navier-Stokes equations will be sooner.
@@Eigensteve 😍😍
❤ thanks.
God bless you ❤️.
I expect a full concert in the next video
Haha yeah… I’m not going to quit my day job…
Nice video and presentation.
Are we talking about mechanical wave or not?
Hi, Steve. Thanks fr the great video. A quiz here, isn't X a function of time?
You are a legend
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!
OMG the guitar is so cool!!
Thanks!! So much fun to play
Very good teaching. From thai student
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.
Bravo👏🏼🎓
Thank you!!
Happy 2024, thank you so much for this excellent lecture. 🎉
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!
maybe c for constant?
@@sorry4all thanks for reply. not sure if c is constant. no what it is or not, what's the physics meaning of it?
@@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.
First, second, third times watching this: _crickets_
Fourth time watching this: "YOOO THAT DERIVATION IS SO COOL!"
crazy good
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!
Thanks! I get it!
super clever :)
I have a series of videos on my channel about deriving the wave equation from Walter Lewin at MIT
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! :)
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.
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.
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.
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.
@@dennisgawera8788 Thank you so much!
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.
Good point. I will be making a video on this soon I hope.
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.
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
i don't see the difference. in either case, you heavily lean on small angles
@@fahrenheit2101 To me that seems a bit tidier and easier to follow. But it assumes that there is no sideways motion of the particles.
High quality
Hi. Can someone explain how the limit becomes the second derivative. I would reallt appreciate this, as then i would fully understand. Lee
Look for the definition of a derivative - it's exactly what we have here.
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.
absolutely true
Is it true for big deflections?
❤️❤️
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?
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).
The answer below is the right idea. We are essentially using the definition of a derivative, which has some assumptions involved.
@@Eigensteve damn, 20 years ago i would have remembered 😭 its all too long ago, but thanks for responding.
LC oscillators in electric circuits too. This is not hyperbole!
😂I love you.
@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!
🥲
What part of this explanation you did not understand when you were a teenager?
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.
i love you
nothing new. it looks like for high school levels
You still need to learn more mathematical physics, some of your fundamentals are still not clear to you 😢