= Solution
The string speed is $c=\sqrt{T/\rho}$. The <characteristic impedance> is the ratio of the forward-wave transverse force to transverse velocity, $Z=T/c=\sqrt{T\rho}$. With peak displacement amplitude $A$, the mean <power> is
$$
\boxed{\langle P\rangle=\tfrac12Z\omega^2A^2}.
$$
Displacement and transverse force are continuous at a massless junction. Since $Z_2=2Z_1$, the displacement amplitudes are $r=(Z_1-Z_2)/(Z_1+Z_2)=-1/3$ and $t=2Z_1/(Z_1+Z_2)=2/3$. Taking the incident amplitude real, the light-string motion is
$$
y=A\left[\cos(kx-\omega t)-\tfrac13\cos(kx+\omega t)\right]
=\frac{2A}{3}\cos(kx-\omega t)+\frac{2A}{3}\sin kx\sin\omega t.
$$
In this <standing-and-travelling decomposition of a reflected string wave>, the first term travels and the second stands; the decomposition is not a separation into independent incident-power fractions because these fields interfere. The reflected wave has a phase reversal. <power> coefficients and standing-wave ratio are
$$
\boxed{\mathcal R=|r|^2=\frac19,\quad
\mathcal T=\frac{Z_2}{Z_1}|t|^2=\frac89,\quad
\mathrm{SWR}=\frac{1+|r|}{1-|r|}=2}.
$$
For a uniform inserted matching section, unequal exterior impedances require a <quarter-wave impedance transformer>: $Z_M=\sqrt{Z_1Z_2}$ and $k_Ma=(2j+1)\pi/2$. At equal <tension> this means \b[middle density $2\rho_0$ and an odd-quarter-wavelength length in that section]. Reflections at its two ends then cancel at the design <frequency>. An arbitrary density or an arbitrary half-wavelength segment cannot perfectly match these unequal exterior strings; the ideal match is frequency-selective.
<amplitude scaling along a slowly tapered string> instead follows from matching over an adiabatic <frequency> range. The source fixes <frequency>, while conserved <power> and the final <characteristic impedance> determine amplitude. In the heavy string,
$$
\boxed{f_2=f_1,\qquad A_2=A_1/\sqrt2,\qquad\lambda_2=\lambda_1/2}.
$$
Its <wave speed> halves. The no-reflection approximation needs small fractional <characteristic impedance> change over each local <wavelength>, rather than length alone for a taper containing abrupt features.
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