Sunligth on the lake bottom

In the water wave number and wave length change according to $k' = n k$ and $\lambda' = \lambda/n$ . Furthermore, absorption causes an exponential decay $E_{(t)} \propto e^{\alpha z}$ of the transmitted beam. The decay coeffizient $\alpha$ is determined by $\frac{I(-d_0)}{I(0)} = e^{- 2\alpha d_0} = \frac{1}{2} \quad \Rightarrow \quad \alpha = \frac{\log 2}{2 d_0} = 0.035 \,\frac{1}{\text{m}}$ Therefore, the electric field can be written as $\begin{aligned} E(z,t) = \begin{cases} E_0 e^{i (\omega t + k z)} + A e^{i (\omega t - k z)} & z > 0 \\ B e^{i (\omega t + n k z)} e^{\alpha z} & z < 0 \end{cases} \end{aligned}$ with unknown coefficients $A$ and $B$ . The connection condition for the water surface results to $\begin{aligned} E(z = 0,t) &= (E_0 + A ) e^{i \omega t} = B e^{i \omega t} \\ E'(z= 0,t) &= i k(E_0 - A ) e^{i \omega t} = (i n k + \alpha) B e^{i \omega t} \approx i n k B e^{i \omega t} \end{aligned}$ In the last line $\alpha$ was neglected, because the eletric field changes only slightly by the absorption on the long scale of a wavelength ( $\lambda \ll d_0$ or $k \gg \alpha$ ). Therefore, a system of linear equations results $\begin{aligned} \left( \begin{array}{cc} -1 & 1 \\ 1 & n \end{array} \right) \left( \begin{array}{c} A \\ B \end{array} \right) &= \left( \begin{array}{c} E_0 \\ E_0 \end{array} \right) \\ \Rightarrow \qquad \left( \begin{array}{c} A \\ B \end{array} \right) &= \frac{-1}{1+n}\left( \begin{array}{cc} n & -1 \\ -1 & -1 \end{array} \right) \left( \begin{array}{c} E_0 \\ E_0 \end{array} \right) = \frac{E_0}{1+n}\left( \begin{array}{c} 1-n \\ 2 \end{array} \right) \end{aligned}$ The light intensity on the lake bottom results $\begin{aligned} I &= \frac{1}{2} c \varepsilon \varepsilon_0 |E(-d)|^2 = \frac{1}{2} n c_0 \varepsilon_0 |B|^2 e^{-2 \alpha d} = \frac{4 n}{(1 + n)^2} e^{-2 \alpha d} I_0 \\ \Rightarrow \quad \frac{I}{I_0} &= \frac{4 n}{(1 + n)^2} e^{-2 \alpha d} \approx 0.741 \end{aligned}$ with the Intensity $I_0 = \frac{1}{2} c_0 \varepsilon_0 |E_0|^2$ .of the incident light.