# Engineering Acoustics/Time-Domain Solutions

## d\'Alembert 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:

$\xi=ct-x\,$ and $\eta=ct+x\,$

and then applying the chain rule to the general form of the wave
equation.

From this, the solution can be written in the form:

$y(\xi,\eta)= f(\xi)+g(\eta)\,=f(x+ct)+g(x-ct)$

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.

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