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Let us consider an incident wave $y_i$ travelling along positive x-axis in medium $1$ which is reflected($y_r$) and refracted(transmitted)($y_t$) across the boundary(at $x=0$) between medium $1$ and another medium $2$.

In this time-stamped video, he explains the reason as to why at $x=0$, $y_i+y_r=y_t$ which implies

$$A_i+A_r=A_t \;\;\;\;\;\;\;\;\;-1$$

However we know that energy of a travlling wave is directly proportional to the square of the amplitude. So one could argue that since $E_i=E_r+E_t$, therefore:

$$A_i^2=A_r^2+A_t^2 \;\;\;\;\;\;\;\;\;-2$$

Clearly $1$ and $2$ are different looking. Is there some flaw in my argument? If no, then are these 2 equations indeed equivalent?

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You do not say what sort of waves these are, but in any case the energy will not just be $A^2$ but $A^2$times some constants $\kappa$ that depend on the material. These will be different in the two media so that $$ A_r+A_t=A $$ is consistent with $$ \kappa_1(A^2_r+A^2_t)=\kappa_2A^2 $$

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  • $\begingroup$ Thanks for the reply! Sorry for explicitly not mentioning that those are travelling transverse waves. So the proportionality constant varies with the nature of media? Thats very interesting! How exactly do they vary(couldnt find any reference)? So is it possible to arrive at $A_i+A_r=A_t$ from the energy equation by manipulating the constants? $\endgroup$
    – DatBoi
    Commented Apr 21, 2021 at 14:37
  • $\begingroup$ Traverse waves on a string,rock, or electromagnetic? If a string than $A_r+A_t=A$ just follows from the string being connected where the dffernt strings join . The energy flux on a string is $-T \dot y y'$ and for a travelling wave the ratio of $\dot y$ to $y'$ depends on the wave speed which is different on the two strings. $\endgroup$
    – mike stone
    Commented Apr 21, 2021 at 14:41
  • $\begingroup$ Thats a wonderful argument with energy fluxes. Now we know that $\dot{y}/y'=-v$. How do I relate $\dot{y}.y'$ with $v$ so that I can compare the constants of the 2 media? $\endgroup$
    – DatBoi
    Commented Apr 21, 2021 at 14:57

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