Q: Do combined waves with non-rational frequencies have a common period? I am facing a problem where I have two waves combined:
\begin{equation}
y = A\sin(b_1x)+B\cos(b_2x)
\end{equation}
Where $ b_1 $ and $ b_2 $ are non-rationals. i.e. \begin{align}
 & b_1 = \sqrt{3+\sqrt{5}}  
&  b_2 = \sqrt{3-\sqrt{5}} \end{align}
I wanted to know if there exists $ X_p $ such that $ y(x+X_p) = y(x) $.
I am confused here because if I actually try to solve this equation, the common procedure is to do the following:
\begin{align}
 & y(x+X_p) = y(x) \\ 
&  A\sin(b_1(x+X_p))+B\cos(b_2(x+X_p)) = A\sin(b_1x+2\pi n)+B\cos(b_2x+2\pi m ) \\
& \forall n,m \in \mathbb{N}  \end{align}
This will give 2 equations in $X_p$ , $n$ and $m$ if the sine and cosine are equated separately. However, I could not see a solution to that and I start to doubt whether a solution exists at all...
EDIT:
I am doubting a solution that can be applied to any A and B.
Thank you!
