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:

${\displaystyle \xi =ct-x\,}$ and ${\displaystyle \eta =ct+x\,}$

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

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

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

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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