Engineering Acoustics/Time-Domain Solutions

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Part 1: Lumped Acoustical Systems1.11.21.31.41.51.61.71.81.91.101.11

Part 2: One-Dimensional Wave Motion2.12.22.3

Part 3: Applications3.13.23.33.43.53.63.73.83.93.103.113.123.133.143.153.163.173.183.193.203.213.223.233.24

d'Alembert Solutions[edit]

In 1747, Jean Le Rond d'Alembertpublished 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[edit]

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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