A transverse wave is described by $y=A\sin (kx-\omega t+\varphi )$. If the wave travels through a medium where the density varies as $\rho (x)={\rho }_0(1+\alpha x)$, what is the change in the wave amplitude as it propagates?

Solution

1. The intensity of a transverse wave is given by $I=\frac{1}{2}\rho {\omega }^2{A}^{2}v$, where $v=\sqrt{T/\rho }$ is the wave speed.
2. Assuming constant power transmission, the intensity $I$ remains constant, implying $I\propto \rho A^{2}v=\text{constant}$.
3. Substituting $v\propto {\rho }^{-1/2}$ into the intensity relation yields $\rho A^2{\rho }^{-1/2}\propto {\rho }^{1/2}{A}^2=\text{constant}$.
4. Solving for amplitude gives $A\propto {\rho }^{-1/4}$, and substituting $\rho (x)={\rho }_0(1+\alpha x)$ results in $A\propto (1+\alpha x{)}^{-1/4}$.
5. Hence the answer is (A).