Function with oscillating frequency?

Question

I'm looking for a function whose frequency oscillates around a certain value (say, oscillating between 440 Hz and 880 Hz, at a rate of 1 Hz -- i.e., its frequency goes up and down once per second, preferably in a somewhat linear fashion).

I feel like it should be quite simple, but I'm having trouble coming up with such a function.

I tried $x(t)=\sin(\sin(t)\ t)$, but this doesn't seem to work, because it's not periodic -- I'm looking for something whose frequency goes up and down in a periodic fashion.

Ideas?

Answer 1

This is frequency modulation, widely used for broadcast radio.

As a simple model, something like $t\mapsto A\sin\big(\omega_1( t + \alpha\sin(\omega_2 t))\big)$ should do the trick, where

• $\omega_1$ is the (angular) frequency of the carrier wave.
• $\omega_2$ is the (angular) frequency of the signal.
• $\alpha$ is a modulation depth that should be kept comfortably below $1$.

If the signal is something more complex than a sine wave, replace $\sin(\omega_2 t)$ by an antiderivative of the signal.