5.5: Exercises

Exercise $\PageIndex{1}$

In Section 5.2, we encountered the complex frequencies $$\omega_\pm = -i\gamma \pm \sqrt{\omega_0^2 - \gamma^2}.$$ For fixed $\omega_0$ and $\omega_0 > \gamma$ (under-damping), prove that $\omega_\pm$ lie along a circular arc in the complex plane.

Exercise $\PageIndex{2}$

Derive the general solution for the critically damped harmonic oscillator, Eq. (5.3.16), by following these steps:

1. Consider the complex ODE, in the under-damped regime $\omega_0 > \gamma$. We saw in Section 5.3 that the general solution has the form $$z(t) = \psi_+ \, \exp\left[\left(-\gamma - i \sqrt{\omega_0^2 - \gamma^2}\right)t\right] \; +\; \psi_- \, \exp\left[\left(-\gamma +i\sqrt{\omega_0^2 - \gamma^2}\right)t\right]$$ for some complex parameters $\psi_+$ and $\psi_-$. Define the positive parameter $\varepsilon = \sqrt{\omega_0^2 - \gamma^2}$. Re-write $z(t)$ in terms of $\gamma$ and $\varepsilon$ (i.e., eliminating $\omega_0$).

2. The expression for $z(t)$ is presently parameterized by the independent parameters $\psi_+$, $\psi_-$, $\varepsilon$, and $\gamma$. We are free to re-define the parameters, by taking $$\begin{align} \alpha &= \psi_+ + \psi_- \\ \beta &= -i\varepsilon(\psi_+ - \psi_-). \end{align}$$ Using these equations, express $z(t)$ using a new set of independent complex parameters, one of which is $\varepsilon$. Explicitly identify the other independent parameters, and state whether they are real or complex.

3. Expand the exponentials in $z(t)$ in terms of the parameter $\varepsilon$. Then show that in the limit $\varepsilon \rightarrow 0$, $z(t)$ reduces to the critically-damped general solution (5.3.16).

Exercise $\PageIndex{3}$

Repeat the above derivation for the critically-damped solution, but starting from the over-damped regime $\gamma > \omega_0$.

Exercise $\PageIndex{4}$

Let $z(t)$ be a complex function of a real input $t$, which obeys the differential equation $$\frac{dz}{dt} = -i\,(\omega_1 - i \gamma)\; z(t),$$ where $\omega_1$ and $\gamma$ are real. Find the general solution for $z(t)$, and hence show that $z(t)$ satisfies the damped oscillator equation $$\left[\frac{d^2}{dt^2} + 2\gamma \frac{d}{dt} + \omega_0^2 \right] z(t) = 0$$ for some $\omega_0^2$. Finally, show that this harmonic oscillator is always under-damped.

Answer

The general solution is $$z(t) = A \exp\left[-i(\omega_1 - i \gamma) t\right].$$ It can be verified by direct substitution that this is a solution to the differential equation. It contains one free parameter, and the differential equation is first-order, so it must be a general solution. Next, $$\begin{align} \frac{d^2z}{dt^2} + 2 \gamma \frac{dz}{dt} &= (-i)^2(\omega_1 - i\gamma)^2 z(t) - 2i \gamma (\omega_1 - i \gamma) z(t) \\ &= \left[- \omega_1^2 + \gamma^2 + 2i\gamma\omega_1 - 2i \gamma \omega_1 - 2\gamma^2)\right] z(t) \\ &= -\left(\omega_1^2 + \gamma^2\right)z(t).\end{align}$$ Hence, $z(t)$ obeys a damped harmonic oscillator equation with $\omega_0^2 = \omega_1^2 + \gamma^2.$ This expression for the natural frequency ensures that $\omega_0^2 > \gamma^2$ (assuming the parameters $\gamma$ and $\omega_1$ are both real); hence, the harmonic oscillator is always under-damped.