Reflection of waves

Wave Equation on Half-Line
In this video, I solve the wave equation on the half line by using the same technique that I used for the heat equation, namely by extending the initial conditions to the whole line and by using D'Alembert's formula. The solution exhibits a cool physical phenomenon: Namely the wave reflects on the axis x = 0 (if time is large enough). Enjoy!
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[@anhquocnguyen1967]

Dr. Peyam, you are the best!

[@dimastus]

Why the reflected wave is not inverted? I think that the whole point of this method is to consider the odd initial condition case, which leads to 4 different waves. Two above the x-axis starting on the right and going in diiferent directions, and two waves on the left side below the x-axis. After some time one left and one right go through each other (canceling out at some moment), but always zero at zero(so the border condition ot the original task is satisfied). Then left wave become right but still below the x-axis, and this is the actual reflected wave.

[@tahseensaad7860]

Yeah, it is a good point. Waves could have three states in what you have had mentioned. But if they reflected in with same magnitude in oposit direction then, (x,t) will be zero! We need to prove the scenario 🤔

[@non-inertialobserver946]

Bruh how is this comment a month old

[@salvatoregiordano9050]

I really love this video and the connection to the application! Thank you!

[@Fortitudeize]

Dr. Peyam, I think the case with two walls shown at the end represents a standing wave. Neat, has application to a infinite potential well in quantum mechanics.

[@jewishjewom12ify]

I’m really glad that you’ve started to more stuff related to your field of research. Any advice or books you recommend for someone who is starting grad school soon and is interested in analysis, PDEs, Fourier series, that sort of thing?

[@drpeyam]

Evans PDE

[@drpeyam]

And anything by Stein and Shakarchi

[@mohammedal-haddad2652]

Brilliant approach. Thank you very much.

[@ellen128]

Great great lecture!!! But isn't (x - ct) for the right-moving wave?

[@drpeyam]

Thank you! And you’re correct

[@armanrasouli2779]

Thanks for the amazing content, Keep up the good job!

[@myrtoh.964]

You are the reason my single last brain cell is still working at 2am studying for my exam. Thank you!

[@MathwithMing]

Awesome stuff

[@dgrandlapinblanc]

Thanks.

[@rodrigo_p2821]

I love your videos!!

[@sanghyeonchang2600]

Sorry, can you explain me why x-ct changes to ct-x at 14:15 ?

[@non-inertialobserver946]

I was just searching for this today, what a coincidence

[@LongNguyen-ub6rz]

Character In the video It's great, I like it a lot $$

[@tahseensaad7860]

Love you dear. 😊😊😊
Really great. One question. Could you share us the book or the method that all of your videos you do? I will be gratefull if you would. 😊

[@drpeyam]

PDEs by Strauss, but I prefer the book by Evans

[@tahseensaad7860]

Thanks alot dear Dr Peyam

[@Gamma_Digamma]

Now I wish I could have been able to solve my PDE in my biophysics project

[@رِهامعدنانحنونأبوالهيل]

sir ..why we using here odd function?

[@drpeyam]

To force the function to be 0 at 0

[@رِهامعدنانحنونأبوالهيل]

@@drpeyam
Thank you ..🌸
sir possibale explain more

[@drpeyam]

If f is odd then f(0) = 0

[@رِهامعدنانحنونأبوالهيل]

thanm you🌸🌸