The Method of Characteristics and the Wave Equation

[@AngeTheGreat]

Great video!

[@Eigensteve]

Thanks!

[@gedaliakoehler6992]

I like both of these guys :)

[@lioneloddo]

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.

[@Shadowkainine]

Perfect timing for the PDE course I'm taking. Love your videos!

[@federicogiargia7648]

Great video! Excellent audio and video quality, and a very clear explanation. Keep up the great work!

[@elsablo2688]

Dear Steve. Thanks for the great video. I think there might be a mistake in the formula for u. It should be 1/2 (f(x+ct)+f(x-ct)), since otherwise the initial condition is not satisfied. Furthermore in the drawing one might be confused that 2 waves of the same height as the initial wave travel in opposite directions, which is not true, since when they meet up again, the overlapping wave would be twice as big as the initial wave. Greetings from Germany!

[@surajsawant5229]

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

[@kaisong3196]

Thank you for this video!!!!!!! I have no idea what's going on with my lecture but this makes so much sense

[@Eigensteve]

Glad it was helpful :)

[@robertburcham7091]

Wonderful lesson, thank you!

[@sdsa007]

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..

[@Eigensteve]

Glad you enjoyed the video!

[@kidslearningbymadhankarthi6114]

Very Nice Video!!!and explained clearly about wave equation solutions!!

This is gold!

[@InusahBojua]

Thank you very much ❤

[@GanyoKumado-w2g]

You are a stone thrown from heaven waooo.You're a nation builder

[@lt4376]

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!

[@mvainio]

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.

[@lt4376]

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!

[@ashh870]

When would get a characteristic equation of u(x,t)=f(x-vt)+g(x-vt)?

[@alienmoonstalker]

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?

[@hoseinzahedifar1562]

Thank you... It was very interesting and very informative...😊😊

[@giovanniminelli5590]

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?

[@muthukamalan.m6316]

Hi Steve, could you help here, where I learn UMAP with proper background

[@Bibi_Mbaape]

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