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The Wave Equation and Slack Line Physics
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- Опубликовано: 5 сен 2024
- This video explores slack line physics as a fun and intuitive example of the wave equation.
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This video was produced at the University of Washington
Thanks so much for these videos Steve, not everyone takes pains to lead a student through all assumptions with care.
Hi Professor Brunton, thanks so much for making these amazing videos! I really appreciate all the hard work you put into teaching, and more importantly, making these videos accessible and free for all. At 05:13, the forcing term coming from the concentrated load (in the middle of the slack line) is introduced. As is, adding the term mgδ(x-x₀) violates the dimensional homogeneity of the wave equation. Here is my thinking:
δ(x-x₀) carries the dimensions of the inverse of its input argument. So if x has the dimensions of length [L], then δ(x-x₀) has the dimension of [L⁻¹].
If we draw out the free body diagram and apply Newton's Second Law in the vertical direction, we get (assuming + direction is upwards):
Tsin(θ+Δθ)-Tsin(θ)-mgδ(x-x₀)Δx = (ρΔx)uₜₜ , where m is the mass of the person standing on the slack line, and ρ is the mass per unit length of the slack line
After some simplification and letting Δx approach zero, we should get:
uₜₜ = c²uₓₓ - (mg/ρ)δ(x-x₀)
As a check, we can examine the dimensions of the forcing term to ensure it has the same dimensions as acceleration [LT⁻²]:
[(mg/ρ)δ(x-x₀)] = ([mg]/[ρ])*[δ(x-x₀)]
[(MLT⁻²)/(ML⁻¹)]*[L⁻¹] = [LT⁻²]
Acc. to my opinion the added term m×g must be divided by rho×L in order to make units consistent. It should also be mentioned that h should be the distance to the center of gravity of the human (who is not a point mass). One may also think about the influence of the horizontal velocity of the human. However, for a human it might be negligible, but, for example, it is not for a cable car.
Very interesting questions. Physical phenomena manifesting in PDEs and PDEs used to model the physical world. Just fascinating stuff!
Great video once more!
Thanks so much for these videos.
I would just like to point out two typos (I think)
- The term you add to the original PDE can not be "m g delta" as its units would be Newtons, but u_{tt} is acceleration.
- The period of the human on the slack line can not be sqrt(g/h) for dimensional reasons too. It should be sqrt(h/g) where h is the height of the centre of mass of the human (I guess).
Congratulations anyway, it is a great series of videos!
A great list of videos. Thanks very much
Great videos, Thank you Steve!
Very well explained. Great video...
super cool thank you! your channel is awesome
“Wop wop wop” frequency 😂
Why the two videos are hidden? Can you please at least mention the title of each video?
Hi Steve!!
you're videos are amazing. could you videos on t-sne or umap either fellow PhD students
If a girl is going down on you and she sucks out your whole soul, is she a compiler then? lul.. Better yet, did you traverse a wormhole?