How do I convert this simple Laplace equation to Z-domain? A basic model of coupled strings (eg. piano) is provided here as:



The principle is that it has two identical string simulations each formed by a delay line and LPF. The outputs of these are summed at the "bridge" in the center. The sum is multiplied by $-H_b(s)$ to get the bridge output, which is then subtracted from each of the string loops to simulate damping from energy lost to the bridge.

The bridge equations are:

$$ V_b(s) = H_b(s) \big( V_1^+(s) + V_2^+(s) \big) $$

$$ H_b(s) \triangleq \frac{2}{2+ R_b(s)/R} $$

Where $R_b(s)$ is the impedance of the bridge and $R$ is the impedance of the string.

So in order to implement this, I need to convert $H_b(s)$ (and/or $V_b(s)$) from a Laplace equation to a Z-domain equation.

I am starting to understand some of the principles of the Laplace and Z-domain but I have never converted anything before and I am not sure how to do this. How I get $H_b(z)$ for this usage?

Once I have $H_b(z)$, do I just multiply it by the sums of the two string outputs and then subtract that result from the string loops?

Also, why is the impedance of the string simply $R$ while the impedance of the bridge is $R_b(s)$? Is impedance of the bridge a Laplace function to imply that its impedance may vary with frequency? (Whereas the string impedance would be frequency independent.) If so, how might it simplify things to treat the bridge impedance as frequency independent too?

Thanks.
I believe, mike, that the answer to your question is likely the Bilinear Transform.  Make sure you identify the significant frequency (likely the resonant frequency of the LPF1 and LPF2 or $H_b(s)$ and apply prewarping to that frequency, so that the digital filter hits at the same place that the analog filter.  Remember the transfer functions of the two delay lines are $z^{-N_1}$ and $z^{-N_2}$.  Or, in the s-domain $e^{-N_1sT}$ and $e^{-N_2sT}$ , where $T$ is the sampling period or the reciprocal of the sample rate.