Finding the times when one cycle occurs for FM wave

[Rick66]

Hello everyone,

I'm trying to find the times when one cycle occurs for a FM wave. For instance, given

y(t) = sin(2∏f$_{c}$*t$_{1}$ + sin(2∏f$_{m}$*t$_{1}$))

at an arbitrary time t$_{1}$, I wish to find the time t$_{2}$ such that

(f$_{c}$* t$_{2}$ + sin(2∏f$_{m}$* t$_{2}$) – (f$_{c}$*t$_{1}$ + sin(2∏f$_{m}$*t$_{1}$) = 1 (cycle).

Now I've tried plotting the wave argument in n-t "space" (i.e. n is cycles),

n = f$_{c}$*t + sin(2∏f$_{m}$*t)

to see if some solution presents itself. For instance, we can see that it gives oscillations about the regularly increasing line n = f$_{c}$*t so that t$_{2}$ ≈1/f$_{c}$ + t$_{1}$ gives an approximate solution. But as to an exact solution, I've hit a brick wall. So if somebody could point me in the right direction it would be greatly appreciated.
Thank you
Rick66

 

[Rick66]

Hello everyone,

I'm trying to find the times when one cycle occurs for a FM wave. For instance, given

y(t) = sin(2∏f$_{c}$*t$_{1}$ + (1/∏)sin(2∏f$_{m}$*t$_{1}$))

at an arbitrary time t$_{1}$, I wish to find the time t$_{2}$ such that

(f$_{c}$* t$_{2}$ + sin(2∏f$_{m}$* t$_{2}$) – (f$_{c}$*t$_{1}$ + sin(2∏f$_{m}$*t$_{1}$) = 1 (cycle).

Now I've tried plotting the wave argument in n-t "space" (i.e. n is cycles),

n = f$_{c}$*t + sin(2∏f$_{m}$*t)

to see if some solution presents itself. For instance, we can see that it gives oscillations about the regularly increasing line n = f$_{c}$*t so that t$_{2}$ ≈1/f$_{c}$ + t$_{1}$ gives an approximate solution. But as to an exact solution, I've hit a brick wall. So if somebody could point me in the right direction it would be greatly appreciated.
Thank you
Rick66

PS: I have made the correction to y(t) above.