Solution to Lossy 1D Wave Equation

$$y(t,x) = e^{-{\left(\mu/2\epsilon \right)}{x/c}} y_r\left(t-{x/c}\right) + e^{{\left(\mu/2\epsilon \right)}{x/c}} y_l\left(t+{x/c}\right)$$

Assumptions:

• Small displacements ($y'\ll 1$)
• Small losses ( $\mu \ll \epsilon \omega$)
• $c \mathrel{\stackrel{\Delta}{=}}\sqrt{K/\epsilon } =$ as before (wave velocity in lossless case)

Components decay exponentially in direction of travel

Sampling with $t = n T$, $x = mX$, and $X=cT$ gives

$$y(t_n,\xm ) = g^{-m} y^{+}(n-m) + g^m y^{-}(n+m)$$

where $g \mathrel{\stackrel{\Delta}{=}}e^{-{\mu T/2\epsilon }}$