Suppose we have a damped harmonic oscillator governed by $$m\ddot x +b \dot x + k x=F(t)$$ where $F$ is the force applied and $x$ is the response (displacement of the mass).
Suppose the force $F=F_1+F_2$ is applied with $F_1=Re(A_1 \exp(i \omega_1 t))$ and $F_2=Re(A_2 \exp(i \omega_2 t))$.
Question: I have to show that the mean power absorbed is $\langle P\rangle=\langle P_1\rangle+\langle P_2 \rangle$, where $\langle P_1 \rangle$ and $\langle P_2 \rangle $ are the powers absorbed if only $F_1$ and $F_2$ were applied, respectively.
My approach: Due to the linearity of the system, the response to $F$ is $x=x_1+x_2$, i.e. the sum of the individual responses to the forces $F_1$ and $F_2$.
The mean energy dissipated is given by $\langle P \rangle =\frac12 b |v|^2$, where $v=\dot x$ ($x$ is complex).
We have $v=\dot x_1+\dot x_2$ so $$\begin{align} \langle P \rangle &=\frac12 b |v|^2 \\ &=\frac 12 b (|\dot x_1|^2+|\dot x_2|^2+2Re(\dot x_1 \dot x_2^*)) \end{align}$$
Now the problem is that I can't get rid of the cross-term. If I could get rid of it, the problem would be solved, I believe.