D'Alembert Solution to the Wave Equation

In this video, we derive the D'Alembert Solution to the wave equation. We use the general solution found in the last couple of videos to solve a Wave PDE problem in an infinite domain with two initial conditions (initial displacement and initial velocity). The resulting solution is the D'Alembert Solution/D'Alembert Formula.

We also show earlier on in the video that the Wave Equation consists of the sum of a forward travelling wave (f(x+ct)) and a backward travelling wave (g(x-ct)), because if we're at the same relative location on the wave (e.g. x+ct/x-ct is a constant), then that relative location has to have a decreasing x (backward travelling) for f(x+ct) and an increasing x (forward travelling) for g(x-ct). I hope my explanation here wasn't too confusing because I feel like that was one of the trickier parts of the video. If you have any questions, let me know!

Questions/requests? Ask me in the comments!

Prereqs: This playlist, especially videos 9-12: https://www.youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD

Lecture Notes: https://drive.google.com/file/d/0BzC45hep01Q4MXVvZWpQVzhYSzQ/view?usp=sharing&resourcekey=0-X0ox_ZTe5my7OJb-LKIhFg
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan/

Special thanks to my Patrons:
- Jennifer Helfman
- Jacob Soares

Видео D'Alembert Solution to the Wave Equation канала Faculty of Khan