07-25-2023-0053 - Intuitive derivation of parametric excitation ; BANDPASS FILTER ; TRIG ID ; SIN ; SOLENOID ; SOIDAL ; SOIDIAL ; DOUBLE DRIVE CROSS PT Z P E ; CP ; ATTENUATION ; CONVOL ; FOURIER ; TRANSFORMS ; DECOUPLED SPECIES ; ETC.. DRAFT

Intuitive derivation of parametric excitation

The above derivation may seem like a mathematical sleight-of-hand, so it may be helpful to give an intuitive derivation. The $q$ equation may be written in the form

${\displaystyle \frac{d^{2}q}{d{t}^2}+{\omega }_n^{2}q=-{\omega }_n^{2}f(t)q}$

which represents a simple harmonic oscillator (or, alternatively, a bandpass filter) being driven by a signal $-{\omega }_n^{2}f(t)q$ that is proportional to its response $q(t)$.

Assume that ${\displaystyle q(t)=A\cos ({\omega }_{p}t)}$ already has an oscillation at frequency ${\omega }_p$ and that the pumping ${\displaystyle f(t)=f_0\sin (2{\omega }_{p}t)}$ has double the frequency and a small amplitude $f_0\ll 1$. Applying a trigonometric identity for products of sinusoids, their product $q(t)f(t)$ produces two driving signals, one at frequency ${\omega }_p$ and the other at frequency $3{\omega }_p$.

${\displaystyle f(t)q(t)=\frac{f_0}{2}A[\sin ({\omega }_{p}t)+\sin (3{\omega }_{p}t)]}$

Being off-resonance, the $3{\omega }_p$ signal is attenuated and can be neglected initially. By contrast, the ${\omega }_p$ signal is on resonance, serves to amplify $q(t)$, and is proportional to the amplitude $A$. Hence, the amplitude of $q(t)$ grows exponentially unless it is initially zero. 

Expressed in Fourier space, the multiplication $f(t)q(t)$ is a convolution of their Fourier transforms $\tilde{F}(\omega )$ and $\tilde{Q}(\omega )$. The positive feedback arises because the $+2{\omega }_p$ component of $f(t)$ converts the $-{\omega }_p$ component of $q(t)$ into a driving signal at $+{\omega }_p$, and vice versa (reverse the signs). This explains why the pumping frequency must be near $2{\omega }_n$, twice the natural frequency of the oscillator. Pumping at a grossly different frequency would not couple (i.e., provide mutual positive feedback) between the $-{\omega }_p$ and $+{\omega }_p$ components of $q(t)$. 

https://en.wikipedia.org/wiki/Parametric_oscillator