Engineering Acoustics/Time-Domain Solutions
In 1747, Jean Le Rond d'Alembert published a solution to the one-dimensional wave equation.
The general solution, now known as the d'Alembert method, can be found by introducing two new variables:
and then applying the chain rule to the general form of the wave equation.
From this, the solution can be written in the form:
where f and g are arbitrary functions, that represent two waves traveling in opposing directions.
A more detailed look into the proof of the d'Alembert solution can be found here.
Example of Time Domain Solution
If f(ct-x) is plotted vs. x for two instants in time, the two waves are the same shape but the second displaced by a distance of c(t2-t1) to the right.
The two arbitrary functions could be determined from initial conditions or boundary values.