Partial Differential Equations 0.15 alpha 

© Leon van Dommelen 

5.2 Comparison with D'Alembert
The example problem of the previous section was also solved in chapter
4 using D'Alembert. It is interesting to compare the
two solutions.
The separation of variables solution took the form:
Some of its nice features are:
 It shows the natural frequencies (tones) to be ,
,...
 It shows the energy, so the strength, of each harmonic.
 The method is not restricted to the onedimensional wave
equation.
The D’Alembert solution took the form
Some of its nice features are
 The pressure can be evaluated at any point without doing infinite
sums.
 It shows how wave fronts propagate.
 It shows regions of influence and dependence.
In short, each method has its advantages and disadvantages.