Right.

> "Such equations can be analysed and solved by means of Laplace transform techniques."

so this part

$ Q(\omega) \ast F(\omega) $

was very straightforward to interpret via application of a Fourier transform instead of a Laplace -- it's a matter of taste which way to go. The convolution is trivial because sine waves turn into delta functions and those are easy to deal with in frequency space.

The wikipedia page on Volterra integral equation indicates that these find application for viscoelastic materials, which incidentally I came across the other day in the context of the Mathieu equation. Much like with sloshing, the induced deformation of the material creates an inertial change in the elastic modulus that requires such a formulation. In other words, the *q(t)* deformation response will have a similar shape as the *h(t)* forcing due to a frequency-dependence of the elastic modulus. With time constants and periods as long as we are dealing with, this can easily occur as the material can "catch up" to the forcing and thus mimic its shape. If the forcing is too fast, the elastic modulus would never have a chance to respond, and so would not be important.

This is what happens with sloshing of the ocean waters as the inertial response due to angular momentum changes (i.e. Chandler wobble) and wind shear (i.e. QBO) induces a similar deformation as a viscoelastic material would show. It makes the math harder of course, but that's how nature works. And dealing with this extra complexity is what computers are good for.

> "Such equations can be analysed and solved by means of Laplace transform techniques."

so this part

$ Q(\omega) \ast F(\omega) $

was very straightforward to interpret via application of a Fourier transform instead of a Laplace -- it's a matter of taste which way to go. The convolution is trivial because sine waves turn into delta functions and those are easy to deal with in frequency space.

The wikipedia page on Volterra integral equation indicates that these find application for viscoelastic materials, which incidentally I came across the other day in the context of the Mathieu equation. Much like with sloshing, the induced deformation of the material creates an inertial change in the elastic modulus that requires such a formulation. In other words, the *q(t)* deformation response will have a similar shape as the *h(t)* forcing due to a frequency-dependence of the elastic modulus. With time constants and periods as long as we are dealing with, this can easily occur as the material can "catch up" to the forcing and thus mimic its shape. If the forcing is too fast, the elastic modulus would never have a chance to respond, and so would not be important.

This is what happens with sloshing of the ocean waters as the inertial response due to angular momentum changes (i.e. Chandler wobble) and wind shear (i.e. QBO) induces a similar deformation as a viscoelastic material would show. It makes the math harder of course, but that's how nature works. And dealing with this extra complexity is what computers are good for.