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:
-
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
\(\omega_0\)
).
-
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.
-
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.