PDE 8 | Wave equation: derivation

An introduction to partial differential equations from a practical viewpoint.
PDE playlist: www.youtube.com...
Part 8 topics:
-- derivation of a model for a vibrating string
-- derivation of wave equation

[@ankurc]

one of the most beautiful mathematics lectures I have ever seen in my life...my heart is jumping with joy

[@Flopsaurus]

To clear something up to others who might have been confused, at 17:13, where he states slope = ux, that x is actually a subscript. He is not multiplying. That's the partial derivative of u with respect to x.

[@MizuHakuryu]

Jesusdragon737 thanks!

[@PurusharthSaxena]

Thanks. Got confused there for a second

[@mohammedouallal2]

Slope is derivative

[@Elianath]

Your approach to teaching this material really helps me understand and accept the assumptions being made to derive the equations. I am having a rough time in my PDE class right now because there is so much math (I am an engineering student) that I just can't symbolically follow everything - let alone remember many of the theorems that I learned way back in Calc 3. Thanks for posting these videos...I hope you will continue to do so!

[@SM_Price]

I went through five different videos before finding this one. Your video made the derivation intuitive and simple.

[@vasanthrajaram1241]

this video was just beautiful.I have many of wave equation derivation video,but none of them gave an intuition to wave equation like this

[@johnholme783]

Thank you for taking the time to produce this video, I now understand my textbook derivation. Simple no nonsense explanation explanation!

[@taojiang1094]

marvelous understanding of PDE, really useful to rookies like me, thanks so much!!!!!

[@LeavingCertMaths]

I have watched the derivation on the mechanics course by MIT. The reason given for constant tension there is that a tiny mass element could be given high acceleration if the 2 tension forces acting on it are unequal. You brought up a few points not mentioned there, so I found your approach helpful.

[@chewie78767]

holy moly the textbook did not make sense at all but your video got me to understand it . THanks

[@ArvindSastry]

Thank you very much! you really explained the wave equation (and the assumptions) very well!

[@LawrenceUdeigwe]

Great lecture !!! One quick question though: If you assumed that theta is close to zero (around 18:40), why would your assumption affect only cos(theta) and not sin(thata). In other words, one could also have assumed that since theta is going to zero, sin(theta) is approximately equal to zero, which will not be good. please help.

[@Flopsaurus]

+Lorens Chuno Approximating a small number with zero is quite different from approximating a number close to 1 with 1. Typically, when theta gets small, sin(theta) is approximated as theta, and cos(theta) is approximated as 1. You can multiply a number by 1 and generally get something close to multiplying it by cos(tiny theta). However, if you multiply a number by 0, you will always get zero, so you will not generally get something close to multiplying it by sin(tiny theta).

[@fredericmoresmau4303]

Why can one always assume that the theta-angle is always small ? Can't there be bigger vibrations happening ? What happens then ?

[@daohung1112]

t iu very short, h iu very small, theta iu very small

[@mohammedouallal2]

Taylor expansion

[@mohammedouallal2]

Sine of theta is approximated to theta (by Taylor expansion) due to very small vibrations. As he approximate cosine of theta to 1. Then, simple trigonometry, theta is equal to u_x

[@Pro-dq9ey]

The 3rd assumption saying it is small vibration, i.e. amplitude of vibration is small, i.e. theta is small angle, in which case sin(theta)≈tan(theta)

[@michellejingdong]

Thank you so much for the video! It explains so clearly! All the assumptions here had been emphasized!

[@ParvaChhantyal]

Hi @16:16, can you explain me how you got second line equation after putting lim h=0 please? bit confused in that one. Thanks

[@kaustubhsinha505]

In differential Calculus, the derivative/ change in a function F(x) to F(x+h) over a given period of time T, is given by the following relation: limit of h tending to 0 d/ dT of F(x) (or rate of change of F(x)) = ( F(x+h) - F(x) )/ h Note that I wrote of as putting brackets would be wrong notation as I cannot represent it in text here. commutant has done the same with the function T here, reducing it to a differential form from a limit change form.

[@HotPepperLala]

f'(x) = lim_h->0 [f(x+h)-f(x)]/h. Let f(x) = g(x)h(x).

[@daohung1112]

+rinwhr hi

[@daohung1112]

+Hung Dao you have any physics maths boong?

[@1aMattes]

i had no pdes yet (bachelor thesis) but still i do need them as a part of electromagn. wave theory, you help me a lot!!

[@roomking-petch8168]

Thanks a lot. Your videos are so great and help clarify many many points.

[@LeavingCertMaths]

There is a derivation here at 50:00, youtube lec 7 vibrations and waves. He says that the tension is the same (to a very good approximation) if the amplitude is small, which I do not understand. I think he mentions the constant string tension idea that I mentioned earlier in another video on masses on pulleys. However, I fully understand your explanation.

[@sova-vlog]

I thought that the external force that you include in the equation of motion should be the weight of the string, mg. However, when h tends to 0, mg tends to 0 either. Thanks for this great lecture prof

[@jolez_4869]

At around 8:30 you say that because the vibrations are small the length of the curve is approximately the same to that of the interwall. However, isn't it true for all vibrations due to assumption 1? It being that all motion is only vertical.

[@MrSakriwedg]

Where I can find the derivation with the x displacement taken into account?

[@mohammedouallal2]

Impressive video! I have one remark. I take it you considered the area of string equals to 1....

[@alokrajawat7619]

Thank you very much! you explained the wave equation very well!

[@kaustubhsinha505]

Halo, at 22:20 just to keep the facts right, one cannot have a completely elastic string which also has a constant mass density (ie. constant T and rho), it is not possible in the physical universe.

[@HotPepperLala]

So the wave equation is just one big assumption..

[@davidsonjoseph8991]

Camel with SunGlasses but it works! 😎

[@amalguptan6716]

Reasonable assumptions

[@wargreymon2024]

It's the best one on the topic so far, but I must say I cannot understand the assumption on small vibration, the whole purpose of wave equation is to understand the vibration, the wave equation should explain large vibration, not neglecting it.

[@tallblondephysicist]

Fantastic video, thanks a lot.

[@LeavingCertMaths]

Perhaps you could expand a bit on the idea of constant tension in a string.

[@diwakarsaran3978]

Thank you very much Sir!!

[@Elianath]

Also, can you recommend any textbooks or materials that presents information the way you do? I have gotten Farlow and Strauss and our class book is Zauderer. I find Zauderer's book to be useless for me. My professor teaches with Strauss basically and that is even hard for me to follow. Farlow is okay sometimes... Thanks again

[@SraigiusLT]

loved it, explained very good. i'm inspired by You,. (Lithuania loves you)

[@rodrigobispo]

Thank you! You explained this very well

[@luisirisarri1085]

But c has not dimensions of velocity ? Thank u very much

[@lukestuemke5445]

When summing the forces in the y (or u) direction, I'm confused why the string's weight was not taken into account. Should we not also be adding -rho(x)hg to the left-hand side? T(x+h,t)sin[theta(x+h, t)] - T(x, t)sin[theta(x,t)] - rho(x)hg = rho(x)hu_tt

[@Michallote]

Yes those are external forces. What would happen is that you would do the limit as usual as dividing by h cancels out for the weight term, and it also doesn't affect the definition of derivative. Meaning you would end up with T u_xx - rho g = rho utt To what u_xx = c^2 utt + c^2 g

[@mrmaaza123]

Great video ! Keep up the good work !

[@Triple9MX]

How do I add damping to this derivation?

[@Chausies7]

Thank you so much Rob!! I love these (and you to a degree...

[@Filip6754]

Doesn anybody know a good source where this derivation is used? I'd like to use it in my thesis, but the teachers aren't quite ready to accept a youtube video as a valid reference.

[@OfficialEnman]

Thank you for making these videos!

[@lcfaskys6711]

Fantastic video, thanks a lot!

[@sunaba2971]

Please can I get a project topic on PDE from you for my Msc project

[@muhaha714]

I just dont get the bit where mass is the density times the length. Isnt density the mass per unit volume? Love the videos btw :D

[@rutger5000]

Yes, but volume isn't necessarily length*width*hight. Volume is depending on your space. In 1 D volume is the same as length, in 2 D volume is the same as surface, in 3 D (real space) volume is what you know it to be. There's volume in all D, but it won't be intuitive anymore. In any case it's very normal to talk about line density (mass per length) or surface density (mass per area)

[@laneellisor7113]

Thank you!

[@davidkwon1872]

Thank you!!!

[@cristophorbutuc-mayer6649]

Hi! quick question.. what's the point of taking c^2 why not just leave it as c?

[@ahmedImam]

see the previos video . PDE 7

[@ericgilkey3549]

From a strictly mathematical perspective, c^2 is technically just some coefficient "a". However, if we want to interpret "a" as a velocity, we need to square it to make the units work out (m^2/s^2). Thus, u_tt (m/s^2) has the same units as c^2*u_xx (m^2/s^2*m/m^2 = m/s^2).

[@cmdrblahdee]

Lol, not used to this notation... was wondering why the slope was U times X for several minutes before I realized you meant the derivative /w respect to X

[@alexleviyev]

You are amazing...

[@daohung1112]

good video. thanks

[@jieunboy]

Thx

[@michaelsun7968]

my mind is blown

[@LeavingCertMaths]

Thanks.

[@panazilian]

thankyou!!

[@skrafi4292]

good thankyou sir

[@veronicapaul6768]

good

[@crystalc1ear]

are you canadian

[@angiefan2481]

Do physicists just get to add assumptions every time the math gets ugly?

[@davidtan8903]

I wish I watched this video when I was a sophomore

[@biohoo22]

has anyone ever been so far as decided what to do more like?

[@Pervertlegenge]

hahaha....this is clever...thanx

[@Postermaestro]

o boi

[@ankurc]

let us assume a spherical cow lol assumptions assumptions and assumptions...but still beautiful.....dont have enough words to thank you

[@scholar1972]

Sir, You tends to be too many assumptions.

[@DelphianSociety]

you don't clear by this video? lmao

[@balaportejean7015]

thx bro Jesus loves you Believe in him and repent bro

[@GeodesicBruh]

Balaporte Jean tf