Writing wave equation for a wave in two different media

Question

Consider a young's double slit set-up in water. Consider a point $p$ on the screen , such that $S_{2}P-S_{1}{P}=s$.
Then , Wave equation for wave from $S_{1}$ at point P: $Asin(2pi/lambda(x) -omega t)$, and for the wave through $S_{2}$: $Asin(2pi/lambda(x+s) -omega t)$=$Asin(2pi/lambda(x) -omega t+ 2pi/lambda(s))$.
Clearly, the phase difference is $2pi/lambda(s)$, where $lambda$ is the wavelength in water.
Now suppose we introduce a glass slab, with refractive index $mu$ in-front of $S_{1}$. The wave for the wave through S2 at point P will be the same. However, How will we write the equation for the wave through $S_{1}$, accounting for the fact that it has travelled through two different media?

Pranay · Answer

The wave through $S_{1}$ has traveled an extra optical path difference of $delta x =  a.(mu_2-mu_1)$ where $a$ is the thickness of the slab, $mu_2$ is the refractive index of the material of slab and refractive index of water is $mu_1$.
Corresponding phase difference is $dfrac{2pi}{lambda}.(delta x)$. where $lambda$ is wavelength of light in vacuum.
hence, equation of light through $S_{1}$ can be given by
$$E=E_osin(omega t-frac{2pi}{lambda}(mu_1x + a.(mu_2-mu_1)))$$
Where $x$ is the geometric distance of point $P$ from $S_1$

Writing wave equation for a wave in two different media

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