# 5.5: Exercises

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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.

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.