Deriving the Wave Equation

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

This video was produced at the University of Washington

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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

[CallOFDutyMVP]

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

[charlesschmidt]

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.

[clairezhao]

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

[paulosimones]

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.

[khanster]

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

[hajsh]

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.

[alilabbene]

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 🙏

[manirarebajeanpaul]

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. 👏

[murillonetoo]

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

[timy]

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.

[danialheidar]

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

[Pier-zlgm]

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 😊

[miroslavvorechovsky]

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.

[shsaa]

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?

[camwhite]

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!

[BioMedUSA]

Played the intro a couple of times - Nice segue!

[liorcohen]

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!

[okhan]

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