I have a misunderstanding concerning what it means for a wave to fully transmit from one medium to another. I will demonstrate. Say we have a wave $\psi_1 =e^{i(kx-\omega t)}$. When it encounters a different medium some returns and some keeps on. Say the amplitude of the reflected wave is $R$ and that of the transmitted wave is $T$. Does full transmission mean $R=0$ or $T=1$? T can generally be $2$ in some cases. Does $R=0 \implies T=1$ but not the reverse or are the two equivalent? I know the energy transmitted is only proportional to $T^2$, but I'm not talking about energy transmission here, only about what are the conditions for a wave to fully go from one medium to another and what it implies. Thank you!
$\begingroup$
$\endgroup$
1
-
2$\begingroup$ In which cases can $T$ be $2$? $\endgroup$– RuslanJan 26, 2021 at 20:26
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
The confusion may have th eorigin in confusing the amplitude coefficients with the intensity coefficients. The amplitude coefficients are ratios between the amplitudes of the waves (incident, reflected, transmited). For sound waves, for examples, these amplitudes can the acoustic pressures. We can call these coefficients R and T. For normal incidence, the boundary condition is indeed T=R+1 and T can be close to 2 if R is close to 1. This happens when there is a very large impedance miss-match between the two media. This is not contradicting anythig because the R and T don't describe energy transfer.
The ratios between intensities at the interface ($R_I$ and $T_I$) satisfy R+T=1 and the energy is conserved. Even though $T_I$ is indeed proportional to T^2, it is not equal to it. The intensity depends both on the amplitude and the impedance of the medium. For plane sound waves is $I=p^2/(2Z)$ where p is the amplitude of the pressure wave and Z is the impedance of the medium. The condition R=1 and T=2 (aproximately) happens when the impedance of the second medium ($Z_2$) is much larger than of the first one ($Z_1$). Like sound going from air to water. You can easily see that if $Z_2>>Z_1$ then the transmitted intensity is negligible even though the amplitude of the transmited wave may be twice the one of the reflected wave.
Now getting to total transmission, this happens at a specific value of the incident angle (intromission angle) and only for specific relationships between the impedances and densities of the two media. When this happens both R and $R_I$ are zero. And the transmission coefficient is $T_I=1$. And so is T. The total transmission does not happen at normal incidence so the relation R+1=T does not apply at the intromissin angle.
