I'm dealing with a few sin waves that represent music. An example is the following
f(x)=sin($\frac{1.47x}{440}$)+sin(x)
How would I represent the time period of this superposed wave? Also if the frequency is possible to find (to my knowledge it isn't), can it be represented with a formula?
$f(t) = \sin( a t ) + \sin (b t) $
We want to find the period $T$ (if it exists) such that
$ f(t + T) = f(t) $
This implies, after some algebraic manipulations that
$ \sin(a t) \cos(aT) + \cos(at) \sin(aT) + \sin(bt) \cos(bT) + \cos(bt) \sin(bT) = \sin(at) + \sin(bt) $
hence,
$ \sin(at ) ( \cos(a T) - 1) + \sin(bt) ( \cos(b T) - 1) + \cos(at) \sin(aT) + \cos(bt) \sin(b T) = 0 $
From the coefficients of $\cos(at)$ and $\cos(bt)$, this implies that
$ a T = k \pi $ and $ b T = m \pi $ for some integers $k$ and $ m$.
since we also want $\cos(aT) = 1$ and $\cos(bT) = 1 $ then $k $ and $m$ must be even integers.
Hence, we want to find even $k$ and $m$ such that
$ T = \dfrac{k \pi }{a} = \dfrac{m \pi}{b} $
The minimum such $T$ is $\pi$ times the least even common multiple of $\dfrac{1}{a}$ and $\dfrac{1}{b} $ , such that the multiples are even multiples.
So for your example function
$ f(x) = \sin(\dfrac{1.47}{440} x) + \sin(x) $
Then
$a = \dfrac{147}{44000} , b = 1 $
And we want the least even common multiple of $\dfrac{44000}{147} , 1 $
Now the least common multiple is $44000$, where $k = 147 $ and $ m = 44000 $. Since we want both $k$ and $m$ to be even, then we have to double $k$ and $m$, so that the least common multiple is $88000$.
Hence, the required period is $88000 \pi$
Plug this in $f(x)$ you get
$ f(x + T) = \sin( \dfrac{1.47}{440} (x + 88000 \pi) ) + \sin( x + 88000 \pi) = \sin( \dfrac{1.47}{440} x + 294 \pi ) + \sin(x) \\ = \sin( \dfrac{1.47}{440} x + 147 (2 \pi) ) + \sin(x)=f(x) $
Note that if we use $T = 44000$ (which is wrong as you'll see), then
$ f(x + T) = \sin( \dfrac{1.47}{440} (x + 44000 \pi) ) + \sin( x + 44000 \pi) = \sin( \dfrac{1.47}{440} x + 147 \pi ) + \sin(x) \\ = \sin( \dfrac{1.47}{440} x + \pi + 73 (2 \pi) ) + \sin(x) \\ - \sin( \dfrac{1.47}{440} x ) + \sin(x) \ne f(x) $