Since I haven’t really touched upon partial differential equations (PDEs) yet in my posts, I thought it would be a good idea to go through one by looking at its derivation and its general solution. As you can probably guess from the title, the PDE that I am going to go through is the 1D wave equation as it is one of the most simplest PDEs but has many important applications in other fields such as physics. The 1D wave equation, shown below, is a second order PDE who’s solutions have wave like properties.
For this post I will just be going through the derivation of this equation and in the next post I will be finding the general solution to the equation using separation of variables.
The Derivation
While I just said that the wave function gives solutions with “wave-like” properties, it’s difficult to define what wave-like really means. It could mean something with periodic properties, or it could mean some disturbance in a medium that can propagate. However, for the sake of this derivation I will be ignoring the nuances like the periodicity or the propagation and will just be interpreting it as something that causes some medium to vibrate since the other details aren’t really required in our derivation. As we are talking about a one dimensional wave equation, we can let our medium just be a vibrating string…
