# Adding waves to a standing wave

#### boks

Joined Oct 10, 2008
218
How can I show that
$$Aexpi(\omega t - kx + \phi_{1}) + Aexpi(\omega t + kx + \phi_{2}) = 2Aexpi(\omega t + \frac{\phi_{1} + \phi_{2}}{2})cos(kx - \frac{\phi_{1} - \phi_{2}}{2})$$?

#### steveb

Joined Jul 3, 2008
2,436
How can I show that
$$Aexpi(\omega t - kx + \phi_{1}) + Aexpi(\omega t + kx + \phi_{2}) = 2Aexpi(\omega t + \frac{\phi_{1} + \phi_{2}}{2})cos(kx - \frac{\phi_{1} - \phi_{2}}{2})$$?
One way is to factor out $$2Aexpi(\omega t + \frac{\phi_{1} + \phi_{2}}{2})$$ from the left hand side.

You will then be left with a much easier problem, which is to show that

$${{expi( - kx + \phi_{1}-\frac{\phi_{1} + \phi_{2}}{2}) + expi( kx + \phi_{2}-\frac{\phi_{1} + \phi_{2}}{2})}\over{2}} = cos(kx - \frac{\phi_{1} - \phi_{2}}{2})$$