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.