Linearity of energy dissipation in a damped harmonic oscillator

Question

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.

Answer 1

So when this happens, the equations are telling you that there is something physical that you are not considering. You can't prove this particular statement because this particular statement is not true in all physical circumstances.

So that should lead you to look at the equations again and see in what limit they might be true, in what limit they might be false.

That counterexample is really easy to generate once you realize that that's what it's telling you, suppose two waves of the same frequency are 180 degrees out of phase. Then the response will be zero and the power it dissipates will be zero: you can add two driving forces that cancel each other out. Or, if they are in phase, the power is now quadruple the original, not double.

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 $1/|\omega_1-\omega_2|.$

On shorter time scales, you might capture the power during a node or a peak of a beat, and this property will not be true.