Root-Power Waves

Wave variables normalized to square root of power carried:

$$\begin{array}{rclrcl} \tilde{f}^{+}&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& f^{{+}}/\sqrt{R}\qquad & \tilde{f}^{-}& \mathrel{\stackrel{\mathrm{\Delta}}{=}}& f^{{-}}/\sqrt{R}\\ \tilde{v}^{+}&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& v^{+}\sqrt{R}\qquad & \tilde{v}^{-}& \mathrel{\stackrel{\mathrm{\Delta}}{=}}& v^{-}\sqrt{R} \end{array}$$

$\Rightarrow$

$$\begin{array}{rcccl} {\cal P}^{+}& = & f^{{+}}v^{+}&=& \tilde{f}^{+}\tilde{v}^{+}\nonumber \\ &=&R\,(v^{+})^2 &=& (\tilde{v}^{+})^2 \\ &=&(f^{{+}})^2 / R&=& (\tilde{f}^{+})^2 \nonumber \end{array}$$

and

$$\begin{array}{rcccl} {\cal P}^{-}& = & -f^{{-}}v^{-}&=& -\tilde{f}^{+}\tilde{v}^{+}\nonumber \\ &=&R\,(v^{-})^2 &=& (\tilde{v}^{-})^2 \\ &=&(f^{{-}})^2 / R&=& (\tilde{f}^{-})^2 \nonumber \end{array}$$

• Normalized wave variables $\tilde{f}^\pm$ and $\tilde{v}^\pm$ behave physically like force and velocity waves
• Either can be squared to obtain signal power
• Dynamic range is normalized in $L_2$ sense
• Driving a normalized waveguide network with unit variance white noise gives signal power equal to $1$ throughout the network
• Time-varying wave impedances do not cause ``parametric amplification''