# Why does phase of a wave change after reflecting from a denser medium?

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

• 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. Commented Apr 16, 2016 at 13:53

## 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.

• 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.
– oops
Commented Apr 16, 2016 at 14:50
• @PrayasAgrawal: Edited. Check now.
– user36790
Commented Apr 16, 2016 at 15:00