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Root-Power Waves

Wave variables normalized to square root of power carried:

\begin{displaymath}
\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}\end{displaymath}

$ \Rightarrow$

\begin{displaymath}
\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}\end{displaymath}

and

\begin{displaymath}
\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}\end{displaymath}


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Download VariableChoice.pdf
Download VariableChoice_2up.pdf
Download VariableChoice_4up.pdf

``Choice of Wave Variables in Digital Waveguide Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2019-02-05 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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