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?