Why is the Laplace Transform essentially never used when dealing with problems involving resonance?

Both the Laplace Transform and the Fourier Transform can be applied to a PDE, for example the wave equation, and used to derive a solution to the equation.

But I never see the Laplace Transform used for problems involving resonance. Such problems always make use of the Fourier Transform. A Google search confirms that the Fourier is much more popular for these problems across all fields.

Why is this, is it not possible to analyse resonant phenomena with the Laplace Transform?

The 'only' difference between the two transforms is that the Laplace Transform makes use of initial conditions and it (in the standard case) is defined on $$t\in [0, \infty)$$. But surely the resonant phenomena is a component of the PDE itself so it should manisfest itself in our analysis whether we use the Laplace Transform or the Fourier Transform?

• Almost everybody in electrical engineering uses Laplace Transform to find resonances by finding the roots of the transfer function's denominator. – hyportnex Oct 26 '18 at 11:51
• Classical control theory makes extensive use of the Laplace transform so I don't believe your Google search confirms that it doesn't. – Alfred Centauri Oct 26 '18 at 12:12
• @AlfredCentauri Google for - "Laplace Transform" resonance - and then Google for - "Fourier Transform" resonance - there are almost 100 times more results for the Fourier case. – csss Oct 27 '18 at 7:09
• @csss, would such a result confirm "that the Fourier is much more popular for these problems across all fields."? Think about the meaning of the word confirm. – Alfred Centauri Oct 27 '18 at 12:19

1 Answer

First I think that usually when solving partial PDEs we consider lossless cases for the ease of calculation. If we consider lossless cases, then there is no need to consider laplace transform (at least in most cases).

Other than that, I think that usually we are only interested in the basic phenomena (say resonance frequency of a cavity). We know that introducing losses, will slightly change resonance frequencies and change Q factors etc, but it is sometimes inpracticable to calculate analytically and we would use simulation tools in order to determine the specific case.

• Ok that makes sense thanks. One other similar issue is that practically all the literature on diffraction of waves in the frequency domain, e.g. geometric/unified theory of diffraction, is given in terms of the Fourier Transform. It's an even more extreme situation than the case of resonance problems. Have you any opinion on why all the frequency domain diffraction literature is focused on the Fourier Transform and not the Laplace Transform? – csss Oct 27 '18 at 7:24