center o bass
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Homework Statement
As in the Title starting from two planewaves with the same amplitude, but different frequency.
Homework Equations
Starting from
Ae^{i(kx +\omega_1 t)} + Ae^{i(kx +\omega_2 )}
The Attempt at a Solution
I got as far as
Ae^{i(kx +\omega_1 t)} + Ae^{i(kx +\omega_2 )}
= 2Ae^{ikx} e^{i \frac{\omega_2 + \omega_1}{2} t} \cos \frac{\omega_2 - \omega_1}2 t
taking the real part
= 2A \left( \cos kx \cos{ \frac{\omega_2 + \omega_1}{2} }t - \sin kx \sin { \frac{\omega_2 + \omega_1}{2} } \right) \cos{ \frac{\omega_2 - \omega_1}{2}}
Now how do i argue to conclude that this is definitley not a standing wave? I know that position and time has to be decoupled, but how can I conclude that they are not?
Can this expression be simplified further?
is it enough to say that this is a product of two factors one which depends only on time and one which does depend on both space and time and therefore can not be decoupled?