= String density ratio for prescribed reflected power
{title2=$\mu_2/\mu_1=[(1+\sqrt{\mathcal R})/(1-\sqrt{\mathcal R})]^{\pm2}$}
For two ideal strings under a common <tension>, the positive <mechanical impedances> satisfy $Z_j=\sqrt{T\mu_j}$, where $\mu_j$ are their <linear mass densities>. The reflected fraction of incident <power> is $\mathcal R=[(Z_1-Z_2)/(Z_1+Z_2)]^2$. Solving for the positive ratio $z=Z_2/Z_1$ gives $z=(1+\sqrt{\mathcal R})/(1-\sqrt{\mathcal R})$ or its reciprocal. Squaring yields the displayed <linear mass density> ratios. For $\mathcal R=1/2$, the heavier string has $17+12\sqrt2\simeq33.97$ times the lighter string's <linear mass density>; the impedance ratio is only $3+2\sqrt2$.
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