Oscillation Frequency

From Fig.F.35, we can see that the impedance of the parallel combination of the mass and spring is given by

$$R_{m\vert\vert k}(s) \isdef \left.\frac{k}{s} \right\Vert ms = \frac{\frac{k}{s}ms}{\frac{k}{s}+ms} = \frac{ks}{s^2+\frac{k}{m}} \protect$$ (F.55)

(using the product-over-sum rule for combining impedances in parallel). The poles of this impedance are given by the roots of the denominator polynomial in $s$ :

$$s = \pm j\sqrt{\frac{k}{m}} \protect$$ (F.56)

The resonance frequency of the mass-spring oscillator is therefore

$$\omega_0 = \sqrt{\frac{k}{m}} \protect$$ (F.57)

Since the poles $s=\pm j\omega_0$ are on the $j\omega$ axis, there is no damping, as we expect.

We can now write reflection coefficient $\rho$ (see Fig.F.37) as

$$\rho = \frac{m-k}{m+k} = \frac{1-\frac{k}{m}}{1+\frac{k}{m}} = \frac{1-\omega_0^2}{1+\omega_0^2}$$

We see that dc ( $\omega_0=0$ ) corresponds to $\rho=1$ , and $\omega_0=\infty$ corresponds to $\rho=-1$ .