A linearly polarised electromagnetic wave given as $\mathbf{E}={\mathrm{E}}_0\hat{\mathrm{i}} \cos (\mathrm{kz}-\mathrm{\omega t})$ is incident normally on a
perfectly reflecting infinite wall at z = a. Assuming that the material of the wall is optically inactive, the
reflected wave will be given as

When a wave is reflected from the denser medium, then the type of wave does not change but only its phase changes by 180⁰ or $\pi$ radian.

Thus, for the reflected wave  $\mathbf{z}=-\mathbf{z},\hat{\mathbf{i}}=-\hat{\mathbf{i}}$  and additional phase of $\pi$ in the incident wave.

Here, the incident EM wave,  $\mathbf{E}=E_0\hat{\mathbf{i}}\cos (kz-\omega t)$

Hence, the reflected EM wave,  ${\mathbf{E}}_r=E_0(-\hat{\mathbf{i}})\cos \left[k\right(-z)-\omega t+\pi ]$

$=-E_0\hat{\mathbf{i}}\cos [-(kz+\omega t)+\pi ]$

$=E_0\hat{\mathbf{i}}\cos (kz+\omega t)$