nad said:

> " I would think that the Mathieu equation would anyways not model the behaviour I was describing, but I would need to look at this longer and I don't have the time for that."

Why wouldn't it model a frequency modulation ?

The Mathieu equation is actually closely related to the generalized wave equation.

Let's look at the network equations for an LC circuit

$ V(t) =- L \frac{di}{dt} $

and,

$ i(t) = C \frac{dV}{dt} $

But now we consider the case that C is variable with time C(t), which places a frequency modulation on the resonance. Substituting

$ V(t) = -L \frac{d}{dt}({C(t) \frac{dV}{dt}} ) $

via the chain rule this will expand into a regular wave equation term plus a modulation that is quite similar to the Mathieu equation.

$ V(t) = -L \frac{dC(t)}{dt} \frac{dV}{dt} - L C(t) \frac{d^2V}{dt^2} $

It is not exactly the same in terms of identical factors (an extra first-order damping term for one) but close enough for understanding.

I would be tempted to use this as a simple model for ENSO sloshing if the literature didn't actually say that Mathieu is the better model.

EDIT: get the signs right

.

> " I would think that the Mathieu equation would anyways not model the behaviour I was describing, but I would need to look at this longer and I don't have the time for that."

Why wouldn't it model a frequency modulation ?

The Mathieu equation is actually closely related to the generalized wave equation.

Let's look at the network equations for an LC circuit

$ V(t) =- L \frac{di}{dt} $

and,

$ i(t) = C \frac{dV}{dt} $

But now we consider the case that C is variable with time C(t), which places a frequency modulation on the resonance. Substituting

$ V(t) = -L \frac{d}{dt}({C(t) \frac{dV}{dt}} ) $

via the chain rule this will expand into a regular wave equation term plus a modulation that is quite similar to the Mathieu equation.

$ V(t) = -L \frac{dC(t)}{dt} \frac{dV}{dt} - L C(t) \frac{d^2V}{dt^2} $

It is not exactly the same in terms of identical factors (an extra first-order damping term for one) but close enough for understanding.

I would be tempted to use this as a simple model for ENSO sloshing if the literature didn't actually say that Mathieu is the better model.

EDIT: get the signs right

.