# Linearity of energy dissipation in a damped harmonic oscillator

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.

So what if they are slightly different frequencies? well we know what that physical phenomenon is, it's called a beat frequency. That beat frequency goes like the difference of the two frequencies, and so this formula can only hold when you are looking at powers over a timescale going like $$1/|\omega_1-\omega_2|.$$