= Reflection at a density step on a string
For a massless join between two ideal strings with a common <tension> $T$, require continuous transverse <displacement> and continuous transverse force $T\partial_x\Psi$. With $Z_j=\sqrt{T\rho_j}$ for <linear mass densities> $\rho_j$, the displacement <reflection coefficient> and transmission amplitude are
$$
r=\frac{Z_1-Z_2}{Z_1+Z_2},\qquad t=\frac{2Z_1}{Z_1+Z_2}.
$$
The reflected and transmitted mean-power fractions are $\mathcal R=|r|^2$ and $\mathcal T=(Z_2/Z_1)|t|^2$, with $\mathcal R+\mathcal T=1$. Reflection from the heavier side reverses <phase>; reflection from the lighter side does not. Total reflection occurs in the limits $Z_2/Z_1\to\infty$ or zero, while a finite positive impedance mismatch transmits some <energy>. Equal <mechanical impedances> remove reflection.
Back to article page
