Uploaded July 2017 | Updated September 2026, 2 weeks ago
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: youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD
Lecture Notes: drive.google.com/file/d/0BzC45hep01Q4MXVvZWpQVzhYSzQ/view?usp=sharing&resourcekey=0-X0ox_ZTe5my7OJb-LKIhFg
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan
Special thanks to my Patrons:
- Jennifer Helfman
- Jacob Soares
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: youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD
Lecture Notes: drive.google.com/file/d/0BzC45hep01Q4MXVvZWpQVzhYSzQ/view?usp=sharing&resourcekey=0-X0ox_ZTe5my7OJb-LKIhFg
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan
Special thanks to my Patrons:
- Jennifer Helfman
- Jacob Soares
