We have solved the wave equation by using Fourier series. While this solution can be derived using Fourier series as well, it is really an awkward use of those concepts.
It is easier and more instructive to derive this solution by making a correct change of variables to get an equation that can be solved by simple integration. We will transform the equation into a simpler form where it can be solved by simple integration.
Therefore, the wave equation 4. Basically, to solve the wave equation or more general hyperbolic equations we find certain characteristic curves along which the equation is really just an ODE, or a pair of ODEs. We know what any solution must look like, but we need to solve for the given side conditions.
We will just give the formula and see that it works. So far so good. And we use the first form of 4.
By the fundamental theorem of calculus we have. We're smoking now. OK, now the boundary conditions.
Consider the simpler setup. It is perhaps easier and more useful to memorize the procedure rather than the formula itself.
The important thing to remember is that a solution to the wave equation is a superposition of two waves traveling in opposite directions.
Linear algebra wave equation pdf
That is,. Let us find the formula again, but slightly differently.
Best approach is to do it in stages. By superposition we get a solution for the general side conditions 4. Check that the new formula 4.
The corners bring us to another interesting remark. One final remark is what the d'Alembert solution tells us about what part of the initial conditions influence the solution at a certain point.
This is why for example we do not know that a supernova has occurred in the universe until we see its light, millions of years from the time when it did in fact happen. Solve using the d'Alembert formula.
The d'Alembert solution still works if there are no boundary conditions and the initial condition is defined on the whole real line. Solve using the d'Alembert solution.
That is, write down a piecewise definition for the solution. Section 4. Named after the French mathematician Jean le Rond d'Alembert —