Wave Digital Mass Derivation

For an ideal mass $m$, we have the driving point impedance

$$R(s) = m s$$

which, when used to terminate a waveguide of impedance $R_0$, gives the reflectance

$$S_m(s) = \frac{m s - R_0}{m s + R_0}$$

(continuous time, Laplace domain). Setting $R_0= m$ gives

$$S_m(s) = \frac{s - 1 }{s + 1}$$

Digitizing using the bilinear transform gives the digital reflectance

$$\tilde{S}_m(z) \isdef S_m\left(\frac{1-z^{-1}}{1+z^{-1}}\right) = -z^{-1}$$

The corresponding difference equation is then simply

$$\zbox{f^{{-}}(n) = - f^{{+}}(n-1)}$$

(wave digital mass).