Lossy Wave Propagation

• Real wave propagation is never lossless. Sound waves in air lose energy via molecular frictional forces. Mechanical vibrations in strings are dissipated through yielding terminations, the viscosity of the surrounding air, and via internal frictional forces. In general, these losses vary with frequency.

• Losses are often well approximated by the addition of a small number of terms to the wave equation.

• In the simplest case, we can add a frequency-independent force term that is proportional to the transverse string velocity. Using the wave equation derived for the string,
  $$T \frac{\partial^{2} y}{\partial x^{2}} = \epsilon \frac{\partial^{2} y}{\partial t^{2}} + \mu \frac{\partial{y}}{\partial{t}},$$
  where $\mu$ is the resistive proportionality constant. Assuming the friction coefficient is relatively small, the following general class of solutions to this equation can be found:
  $$y(t, x) = e^{-(\mu/2\epsilon)x/c}y^{+}(t - x/c) + e^{(\mu/2\epsilon)x/c}y^{-}(t + x/c).$$
  When this solution is sampled, we get
  $$y(t_n, x_m) = g^{m}y^{+}(n - m) + g^{-m}y^{-}(n + m),$$
  where $g = e^{-\mu T_s/2\epsilon}$.

  Figure 7: Discrete-time simulation of ideal, lossy wave propagation.

  

• Because the system is linear and time-invariant, the loss terms can be commuted and implemented at discrete points for efficiency.

• In the more realistic situation where losses are frequency dependent (and typically of “lowpass” characteristic), the $g$ factors are replaced with frequency responses of the form $G(\omega)$. These responses can likewise be commuted and implemented at discrete spatial locations within the system.