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Limiting Cases

Let's consider a couple of "limiting cases" of such solutions. First, suppose that the linear restoring force is extremely weak compared to the "drag" force - i.e.^13.7   $\kappa \gg \omega = \sqrt{k \over m}$. Then   $\sqrt{ \kappa^2 - 4 \omega^2 } \approx \kappa$  and the solutions are   $K \approx 0$ [i.e.  $x \approx$ constant, plausible only if  x = 0] and   $K \approx - \kappa$,  which gives the same sort of damped behaviour as if there were no restoring force, which is what we expected.

Now consider the case where the linear restoring force is very strong and the "drag" force extremely weak - i.e.   $\kappa \ll \omega = \sqrt{k \over m}$. Then   $\sqrt{ \kappa^2 - 4 \omega^2 } \approx 2 \, i \, \omega$  and the solutions are   $K \approx - \onehalf \kappa \pm i \, \omega$,  or^13.8

$$x_0 \; e^K$$ x(t) = (13.23)

$$\approx x_0 \; \exp( \pm i \, \omega t \; - \gamma t )$$ (13.24)

$$x_0 \; e^{\pm i \, \omega t} \cdot e^{- \gamma t}$$ = (13.25)

where   $\gamma \equiv \onehalf \kappa$. We may then think of  $[i \, K]$  as a complex frequency^13.9 whose real part is   $\pm \omega$  and whose imaginary part is  $\gamma$. What sort of situation does this describe? It describes a weakly damped harmonic motion in which the usual sinusoidal pattern damps away within an "envelope" whose shape is that of an exponential decay. A typical case is shown in Fig. 13.5.

  

Figure: Weakly damped harmonic motion. The initial amplitude of  x  (whatever  x  is) is  x₀, the angular frequency is  $\omega$ and the damping rate is  $\gamma$. The cosine-like oscillation, equivalent to the real part of   $x_0 \, e^{i \, \omega \, t}$,  decays within the envelope function   $x_0 \, e^{- \gamma \, t}$.