= Quarter-wave impedance transformer
{wiki}
A lossless intermediate section of positive real <characteristic impedance> $Z_M$ matches unequal real exterior <characteristic impedances> $Z_L,Z_R$ when $Z_M^2=Z_LZ_R$ and its propagation <wave phase> is an odd multiple of $\pi/2$. The length is an odd quarter of the internal <wavelength>. This principle also applies to the <characteristic impedance of a stretched string>.
For a uniform intermediate string of length $l$, put $\delta=k_Ml$. Continuity of displacement and transverse <force> gives the displacement <reflection coefficient>
$$
r=\frac{(Z_L-Z_R)\cos\delta+i(Z_M-Z_LZ_R/Z_M)\sin\delta}{(Z_L+Z_R)\cos\delta-i(Z_M+Z_LZ_R/Z_M)\sin\delta}.
$$
At real positive <mechanical impedances>, the denominator never vanishes. Its numerator vanishes only if $(Z_L-Z_R)\cos\delta=0$ and $(Z_M^2-Z_LZ_R)\sin\delta=0$. Thus unequal exterior strings require the quarter-wave condition. Equal exterior strings with a different middle string instead transmit perfectly at integer half-<wavelengths>; if all three <mechanical impedances> agree, every <frequency> is matched.
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