The Method of Characteristics and the Wave Equation
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- Опубликовано: 24 июл 2024
- Here we discuss the Method of Characteristics, which is a powerful technique to analyze the wave equation. This is used throughout physics, for example to analyze shock waves in compressible fluid dynamics.
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This video was produced at the University of Washington
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0:00 Overview and Recap
3:19 Showing f(x+ct) and f(x-ct) are Solutions
7:07 Example of Traveling Wave
10:54 Changing the Boundary Conditions: Reflecting BCs
14:14 Revisiting the Guitar String 
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Perfect timing for the PDE course I'm taking. Love your videos!
For me, what it was surprinsing was to see that when we multiply sin(pi.x/L) by sin(2.pi.x/L) and add all the values for all the x between 0 and L, it is eqal to 0. It's at this astonishing moment that I have really understood what it is the concept of orthogonality !!!
The concept of orthogonality can be intuitively understood by just looking at a corner in a room.
But thanks to this, we have a deeper undestranding, a more profonfond comprehension of the concept of orthogonality.
This concept is one of the most beautiful concept in Physics. Between a corner in a room and the sound of a guitar string, there is an invisible link : the orthogonality. Maths show us harmonies in the physical word that we can't even imagine !
If linear algebra is a so powerful tool for describing the physical world is also due to this wonderful concept of orthogonality.
Very Nice Video!!!and explained clearly about wave equation solutions!!
very good explanation Sir, I watched a video from an IIT college lecture being Indian ofcourse but didn't understand but your way of teaching with example of the guitar string made me visualise what the equation is trying to say of the string, thankyou
Wonderful lesson, thank you!
Great video!
Thanks!
I like both of these guys :)
Thank you... It was very interesting and very informative...😊😊
This is gold!
Thank you very much ❤
Great video, I really got a new perspective with the 'two-wave equation', shock-waves, traveling waves, super-positioning, and information speeds Thanks! In the past it was taught to me as as the most simplest PDE, and I didn't get the nuance of what it represents physically. I really liked that you contrasted these concepts with the elliptical equations esp the heat equation, which I just finished studying! My initial motivation for studying is to understand Schrodingers equation for QM.... but I am taking the scenic route to imaginary planes..
Glad you enjoyed the video!
This is excellent! If you extend to 3D, please, do the acoustic waves in our vocal tracts. Speech acoustics will have different boundary conditions and the wave is longitudinal; i.e., two new interesting features to study.
Hi professor, love your videos! If I may ask: Steve Mould recently published a video about the effect of a vibrating square, how do you formulate such effect?
Nice video. Could you briefly touch on dispersion, where for some nonlinear systems the wave speed is a function of frequency? Can you model the guitar string without the small angle approximation to keep the nonlinearities (or does it not buy you anything in terms usefulness?
Hi Steve,
could you help here, where I learn UMAP with proper background
When would get a characteristic equation of u(x,t)=f(x-vt)+g(x-vt)?
9:50-9:55 ‘the initial condition disturbance [wave] propagates out at speed c’. Can you show how/when there are waves that travel faster than wave speed by manipulating the original EOM you started with on upper LHS? Thanks!
Aren’t incompressible flows elliptical PDEs? I would wonder how a pressure wave would travel through that. Given that it’s instant information transfer from perspective of the math PDE equation, but not really so for the physics. That’s difficult to understand. Thanks for your lectures helps!
You are a stone thrown from heaven waooo.You're a nation builder
10:00 that 3d graph you made should have u(x,y) as the vertical line and t as the line that extends though diagonally