A transverse wave is described by $y=A\sin (kx-\omega t)$. If the wave travels from a medium with linear mass density ${\mu }_1$ to a medium with linear mass density ${\mu }_2$, what is the ratio of the amplitude of the reflected wave to the incident wave?

Solution

1. At the boundary between two media, the displacement and the transverse force must be continuous, leading to the reflection coefficient $R=\frac{v_2-v_1}{v_2+v_1}$.
2. The wave speed in a string is given by $v=\sqrt{\frac{T}{\mu }}$, where $T$ is the tension and $\mu$ is the linear mass density.
3. Substituting $v_1=\sqrt{\frac{T}{{\mu }_1}}$ and $v_2=\sqrt{\frac{T}{{\mu }_2}}$ into the reflection coefficient formula yields $R=\frac{\sqrt{T/{\mu }_2}-\sqrt{T/{\mu }_1}}{\sqrt{T/{\mu }_2}+\sqrt{T/{\mu }_1}}$.
4. Simplifying the expression by canceling the common tension term $T$ results in $R=\frac{\sqrt{{\mu }_1}-\sqrt{{\mu }_2}}{\sqrt{{\mu }_1}+\sqrt{{\mu }_2}}$.
5. Hence the answer is (A).