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(A) The intensity $I$ of a travelling wave is given by $I = \frac{1}{2} \rho \omega^2 A^2 v$,where $\rho$ is the density of the medium,$\omega$ is the

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$\frac{I_i}{I_r} = \left( \frac{a_i}{a_r} \right)^2$

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(A) The intensity $I$ of a travelling wave is given by $I = \frac{1}{2} \rho \omega^2 A^2 v$,where $\rho$ is the density of the medium,$\omega$ is the angular frequency,$A$ is the amplitude,and $v$ is the wave speed.For the incident and reflected waves,they travel in the same medium,so $\rho$,$\omega$,and $v$ are identical for both. Thus,the ratio of intensities is proportional to the square of the ratio of amplitudes: $\frac{I_i}{I_r} = \frac{\frac{1}{2} \rho \omega^2 a_i^2 v}{\frac{1}{2} \rho \omega^2 a_r^2 v} = \left( \frac{a_i}{a_r} \right)^2$.Since the transmitted wave travels into a different medium,the density $\rho$ and wave speed $v$ change,meaning the relationship between $I_t$ and $a_t$ involves different constants compared to the incident wave. Therefore,only the ratio involving the incident and reflected waves in the same medium holds as $\frac{I_i}{I_r} = \left( \frac{a_i}{a_r} \right)^2$.

$A$ travelling wave is partly reflected and partly transmitted from a rigid boundary. Let $a_i, a_r$ and $a_t$ be the amplitudes of the incident wave,reflected wave,and transmitted wave,and $I_i, I_r$ and $I_t$ be the corresponding intensities. Choose the correct alternative.

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