Engineering Acoustics/Time-Domain Solutions

Part 1: Lumped Acoustical Systems – 1.1 – 1.2 – 1.3 – 1.4 – 1.5 – 1.6 – 1.7 – 1.8 – 1.9 – 1.10 – 1.11

Part 2: One-Dimensional Wave Motion – 2.1 – 2.2 – 2.3

Part 3: Applications – 3.1 – 3.2 – 3.3 – 3.4 – 3.5 – 3.6 – 3.7 – 3.8 – 3.9 – 3.10 – 3.11 – 3.12 – 3.13 – 3.14 – 3.15 – 3.16 – 3.17 – 3.18 – 3.19 – 3.20 – 3.21 – 3.22 – 3.23 – 3.24

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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