Sum of incident and reflected waves being the transmitted waves amplitude

Question

Suppose we have an incident wave $\psi_i$ heading towards an interface, then when it reaches the interface it is split into a reflected and a transmitted wave:

$$ \psi_i \to \psi_r + \psi_t $$

Why then do we calculate the reflection and transmission by saying that the sum of the amplitudes of the incident and reflected waves equals the amplitude of the transmitted wave?

$$ \psi_i + \psi_r = \psi_t $$

Surely the sum of the reflected wave and the transmitted wave should equal the amplitude of the incident wave since the incident wave is broken down into these two elements.

Answer 1

You seem to assume that there exists some sort of "conservation of the amplitude" for a wave. Such a law doesn't exist.

Also, be careful about each function's validity domain. Let's say that the incident and reflected waves live in $x<0$ while the transmitted wave lives in $x>0$. Since those domains don't overlap (except maybe on $x=0$), adding $\psi_r$ and $\psi_t$ doesn't make any sense.

The only physical law that you can write with any generality here is the continuity of the total amplitude:

• It's trivially continuous on $x<0$ and $x>0$.
• The only non-trivial part is on $x=0$.

On $x=0^-$ (left-side limit), total amplitude is $\psi_i+\psi_r$, while on $x=0^+$, total amplitude is only $\psi_t$. Hence the relation: $$\psi_i(0^-)+\psi_r(0^-)=\psi_t(0^+)$$