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Why does phase of a wave change when reflected from a denser medium, but no change takes place in phase during transmission?

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  • $\begingroup$ No time to write a proper answer now, but think about the common intro physics demos involving waves travelling on a rope. This is like reflection from a tethered rope end vs reflection from a free rope end. $\endgroup$ Commented Apr 16, 2016 at 13:53

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tl;dr:

It's because of discontinuous change in impedance $\rm Z.$

Reflection Coefficient $\rm R$:

Let the point where two media meet be $z=0\;.$ At $z= 0\,,$ the incident wave is given as $$\psi_\textrm{inc}(0,t)= A\cos\omega t\;.$$

The total force exerted by the input terminal of the second medium is given by $$\begin{align}F&= \underbrace{F_\textrm{term}}_\textrm{termination force exerted for termination of the incident wave if the second media had the same impedance as the first one}+ \underbrace{F_\textrm{exc}}_\textrm{excess force exerted due to difference in media}\\ &= -\mathrm Z_1\frac{\partial \psi_\textrm{inc}(0,t)}{\partial t}+ \mathrm Z_1\frac{\partial \psi_\textrm{ref}(0,t)}{\partial t}\tag 1\end{align}$$

But the force is equivalent to the force provided by a damper of the same impedance; then $F$ is given by $$F=-\mathrm Z_2\frac{\partial \psi_\textrm{inc}(0,t)}{\partial t} -\mathrm Z_2\frac{\partial \psi_\textrm{ref}(0,t)}{\partial t}\tag 2$$

Equating $(1)$ and $(2)\,,$ we get

$$\frac{\partial \psi_\textrm{ref}(0,t)}{\partial t}= \underbrace{\left[\frac{\mathrm Z_1- \mathrm Z_2}{\mathrm Z_1+ \mathrm Z_2}\right]}_\textrm{reflection coefficient}\frac{\partial \psi_\textrm{inc}(0,t)}{\partial t}$$

The term in the parenthesis is termed by $\rm R_{12}\,,$ the reflection coefficient.

The total displacement $\psi(z,t)$ is given by

\begin{align}\psi(z,t)&= \psi_\textrm{inc}(z,t) + \psi_\textrm{ref}(z,t)\\ &= A\cos(\omega t- k_1z) + \mathrm R_{12}A\cos(\omega t+ k_1z)\tag 3\end{align}

Transmission coefficient $\rm T$:

Let the transmitted wave be given as $$\psi_\textrm{trans}(z,t)= (\mathrm T\,A)\,\cos(\omega t- k_2z)\tag 4$$

Since, the wavefunction must be smooth i.e. continuous at the boundary; equating $(3)$ and $(4)\,$ we get the amplitude transmission coefficient $\rm T$ as $$\rm T= 1+ R \;.$$

Why does phase of a wave change when reflected from a denser medium, but no change takes place in phase during transmission?

When a wave goes from a medium of lower impedance (that is low density) to a medium of higher impedance (that is high density) viz. $\rm Z_2 \gt Z_1\,$ then $\rm R_{12}\lt 0$ and that flips the sign of the wave.

Also, then $\rm T\in (0, 1)$ and that would mean the transmitted wave has the same sign as that of the incident wave; that is it doesn't flip.

Edit in response to OP's comment:

why does phase of a wave doesnt change even if its reflected from a rarer medium

Because now, $\rm Z_1\gt Z_2 $ which implies $\rm R_{12}\gt 0$ and thus $\psi_\textrm{ref}(z,t)$ remains same; its phase doesn't change.

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  • $\begingroup$ But, why does phase of a wave doesnt change even if its reflected from a rarer medium(i.e back reflection of light wave when passing from optically denser to rarer medium. $\endgroup$
    – oops
    Commented Apr 16, 2016 at 14:50
  • $\begingroup$ @PrayasAgrawal: Edited. Check now. $\endgroup$
    – user36790
    Commented Apr 16, 2016 at 15:00

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