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I hope the experts of the field forgive me for this n00b questions, but I am just trying to understand physics. Assume the following function:

$$\phi(t)=\omega t+\cos(\omega t)$$

The above function has the property that:

$$\phi(t+T)-\phi(t)=\omega T=const.$$ Where $T=\frac{2\pi}{\omega}$.

Now the question is based on these facts, may I consider that $\phi$ actually represents the phase of an oscillator??

NOTE: I know my question is a bit silly because I am not modelling any kind of real physical phenomenon but only introducing a function, but I imagine the function I have written is simple case of an oscillator perturbed by means of a second oscillator.

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3 Answers 3

up vote 1 down vote accepted

You can have any function be the phase of an oscillator. Whatever $f(t)$ is, you can speak of $\cos (f(t))$ Your particular expression would make particular sense for phase modulation of a wave. The carrier wave is at frequency $\omega$. Normally the phase modulation would have smaller amplitude than $1$ and probably a lower frequency than $\omega$, but we can plot $\cos(x+\cos(x))$ and get something that is nicely periodic, though distorted from a standard cosine wave.

enter image description here

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Thank you very much. I have additional questions. By the amplitude of one, you are talking about the coefficient of $\cos(\omega) t$ in my expression of phase, right? And my second question: So any mapping (as you applied $\cos$ here) would make sense? the only important thing is something to have the same period with the main phase, right? –  Cupitor Mar 13 at 21:48
1  
Yes, the amplitude of 1 is the range of the inner cosine function. If you want it to be truly periodic, you need the modulation to have a period that is (a multiple of) the period of the main wave. But if you think of phase modulating a data transmission the function would be $\cos (\omega t + data)$ where data is not periodic. The resulting wave is not truly periodic, but it is close enough for many purposes to be considered periodic. –  Ross Millikan Mar 13 at 22:01
    
Thank you very much. –  Cupitor Mar 13 at 22:46

This sort of thing actually shows up both in the engineering aspects of FM radio, as well as in semiclassical representations of intermode coupling in molecular Raman scattering.

Since $\phi$ is periodic, trigonometric functions of $\phi$ such as $\cos(\omega t+\cos(\omega t))$ admit a Fourier series expansion obtainable by the Jacobi-Anger identity. The sidebands are used in FM to transmit information, and in Raman scattering they represent the interaction between two oscillators.

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For the future readers (if any), I highly suggest the Synchronization book. In page 31 of the book there is a thorough discussion of the phase. I am not quite sure writing the books material would be a copyright infringement but I highly recommend it to the ones who are curious to learn more.

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